Combined Feed
https://feeds.library.caltech.edu/people/Schulman-L-J/combined.rss
A Caltech Library Repository Feedhttp://www.rssboard.org/rss-specificationpython-feedgenenTue, 16 Apr 2024 15:57:51 +0000The maintenance of common data in a distributed system
https://resolver.caltech.edu/CaltechAUTHORS:AWEfocs91
Authors: {'items': [{'id': 'Awerbuch-B', 'name': {'family': 'Awerbuch', 'given': 'Baruch'}}, {'id': 'Schulman-L-J', 'name': {'family': 'Schulman', 'given': 'Leonard J.'}, 'orcid': '0000-0001-9901-2797'}]}
Year: 1991
DOI: 10.1109/SFCS.1991.185413
A basic task in distributed computation is the maintenance at each processor of the network, of a current and accurate copy of a common database. A primary example is the maintenance, for routing and other purposes, of a record of the current topology of the system.
Such a database must be updated in the wake of locally generated changes to its contents. Due to previous disconnections of parts of the network, a maintenance protocol may need to update processors holding widely varying versions of the database.
We provide a deterministic protocol for this problem, which has only polylogarithmic overhead in its time and communication complexities. Previous deterministic solutions required polynomial overhead in at least one of these measures.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/nd79v-twd14Communication on noisy channels: a coding theorem for computation
https://resolver.caltech.edu/CaltechAUTHORS:20120328-141319570
Authors: {'items': [{'id': 'Schulman-L-J', 'name': {'family': 'Schulman', 'given': 'Leonard J.'}, 'orcid': '0000-0001-9901-2797'}]}
Year: 1992
DOI: 10.1109/SFCS.1992.267778
Communication is critical to distributed computing, parallel computing, or any situation in which automata interact-hence its significance as a resource in computation. In view of the likelihood of errors occurring in a lengthy interaction, it is desirable to incorporate this possibility in the model of communication. The author relates the noisy channel and the standard (noise less channel) complexities of a communication problem by establishing a `two-way' or interactive analogue of Shanon's coding theorem: every noiseless channel protocol can be simulated by a private-coin noisy channel protocol whose time bound is proportional to the original (noiseless) time bound and inversely proportional to the capacity of the channel, while the protocol errs with vanishing probability. The method involves simulating the original protocol while implementing a hierarchical system of progress checks which ensure that errors of any magnitude in the simulation are, with high probability, rapidly eliminated.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/emcss-gjm22Signal Propagation, with Application to a Lower Bound on the Depth of Noisy Formulas
https://resolver.caltech.edu/CaltechAUTHORS:20120309-133211732
Authors: {'items': [{'id': 'Evans-W', 'name': {'family': 'Evans', 'given': 'Williams'}}, {'id': 'Schulman-L-J', 'name': {'family': 'Schulman', 'given': 'Leonard J.'}, 'orcid': '0000-0001-9901-2797'}]}
Year: 1993
DOI: 10.1109/SFCS.1993.366827
We study the decay of an information signal propagating through a series of noisy channels. We obtain exact bounds on such decay, and as a result provide a new lower bound on the depth of formulas with noisy components. This improves upon previous work of N. Pippenger and significantly decreases the gap between his lower bound and the classical upper bound of von Neumann. We also discuss connections between our work and the study of mixing rates of Markov chains.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/wpz9a-b4443Coding for distributed computation
https://resolver.caltech.edu/CaltechAUTHORS:20120223-084134446
Authors: {'items': [{'id': 'Schulman-L-J', 'name': {'family': 'Schulman', 'given': 'Leonard J.'}, 'orcid': '0000-0001-9901-2797'}]}
Year: 1994
DOI: 10.1109/WITS.1994.513866
The author describes analogous coding theorems for the more general, interactive, communications required in computation. In this case the bits transmitted in the protocol are not known to the processors in advance but are determined dynamically. First he shows that any interactive protocol of length T between two processors connected by a noiseless channel can be simulated, if the channel is noisy (a binary symmetric channel of capacity C), in time proportional to T 1/C, and with error probability exponentially small in T. He then shows that this result can be extended to arbitrary distributed network protocols. He shows that any distributed protocol which runs in time T on a network of degree d having noiseless communication channels, can, if the channels are in fact noisy, be simulated on that network in time proportional to T 1/C log d. The probability of failure of the protocol is exponentially small in T.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/5gv8m-ed141Information Theory and Noisy Computation
https://resolver.caltech.edu/CaltechAUTHORS:20120216-114451513
Authors: {'items': [{'id': 'Evans-W-S', 'name': {'family': 'Evans', 'given': 'William S.'}}, {'id': 'Schulman-L-J', 'name': {'family': 'Schulman', 'given': 'Leonard J.'}, 'orcid': '0000-0001-9901-2797'}]}
Year: 1995
DOI: 10.1109/ISIT.1995.550443
We report on two types of results. The first is a study of the rate of decay of information carried by a signal which is being propagated over a noisy channel. The second is a series of lower bounds on the depth, size, and component reliability of noisy logic circuits which are required to compute some function reliably. The arguments used for the circuit results are information-theoretic, and in particular, the signal decay result is essential to the depth lower bound. Our first result can be viewed as a quantified version of the data processing lemma, for the case of Boolean random variables.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/612e5-wk075Coding for interactive communication
https://resolver.caltech.edu/CaltechAUTHORS:20120224-084221884
Authors: {'items': [{'id': 'Schulman-L-J', 'name': {'family': 'Schulman', 'given': 'Leonard J.'}, 'orcid': '0000-0001-9901-2797'}]}
Year: 1995
DOI: 10.1109/ISIT.1995.550439
Let the input to a computation problem be split between two processors connected by a communication link; and let an interactive protocol π be known, by which on any input, the processors can solve the problem using no more than T transmissions of bits between them, provided the channel is noiseless. We study the following question: if in fact there is some noise on the channel, what is the effect upon the number of transmissions needed in order to solve the communication problem reliably?.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/2gn4j-keh80Coding for Distributed Computation
https://resolver.caltech.edu/CaltechAUTHORS:20120224-093023413
Authors: {'items': [{'id': 'Rajagopalan-S', 'name': {'family': 'Rajagopalan', 'given': 'Sridhar'}}, {'id': 'Schulman-L-J', 'name': {'family': 'Schulman', 'given': 'Leonard J.'}, 'orcid': '0000-0001-9901-2797'}]}
Year: 1995
DOI: 10.1109/ISIT.1995.550440
We show that any distributed protocol which runs on a noiseless network in time T, can be simulated on an identical noisy network with a slow-down factor proportional to log(d+1), where d is the maximum degree in the network, and with exponentially small probability of error. On every channel of our network we implement communications using tree codes.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/57tqs-xjg08Splitters and near-optimal derandomization
https://resolver.caltech.edu/CaltechAUTHORS:20120223-112750729
Authors: {'items': [{'id': 'Naor-M', 'name': {'family': 'Naor', 'given': 'Moni'}}, {'id': 'Schulman-L-J', 'name': {'family': 'Schulman', 'given': 'Leonard J.'}, 'orcid': '0000-0001-9901-2797'}, {'id': 'Srinivasan-A', 'name': {'family': 'Srinivasan', 'given': 'Aravind'}}]}
Year: 1995
DOI: 10.1109/SFCS.1995.492475
We present a fairly general method for finding deterministic constructions obeying what we call k-restrictions; this yields structures of size not much larger than the probabilistic bound. The structures constructed by our method include (n,k)-universal sets (a collection of binary vectors of length n such that for any subset of size k of the indices, all 2^k configurations appear) and families of perfect hash functions. The near-optimal constructions of these objects imply the very efficient derandomization of algorithms in learning, of fixed-subgraph finding algorithms, and of near optimal ΣIIΣ threshold formulae. In addition, they derandomize the reduction showing the hardness of approximation of set cover. They also yield deterministic constructions for a local-coloring protocol, and for exhaustive testing of circuits.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/wq91j-3w168Verifying Identities
https://resolver.caltech.edu/CaltechAUTHORS:20120208-152850561
Authors: {'items': [{'id': 'Rajagopalan-S', 'name': {'family': 'Rajagopalan', 'given': 'Sridhar'}}, {'id': 'Schulman-L-J', 'name': {'family': 'Schulman', 'given': 'Leonard J.'}, 'orcid': '0000-0001-9901-2797'}]}
Year: 1996
DOI: 10.1109/SFCS.1996.548520
We provide an O˜(n^2) time randomized algorithm to check whether a given operation f:S×S→S is associative (letting n=|S|). They prove this performance is optimal (up to polylogarithmic factors) even in case the operation is "cancellative". No sub-n^3 algorithm was previously known for this task. More generally they give an O(n^c ) time randomized algorithm to check whether a collection of c-ary operations satisfy any given "read-once" identity.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/yx1q3-1mz64Coding for interactive communication
https://resolver.caltech.edu/CaltechAUTHORS:SCHUieeetit96
Authors: {'items': [{'id': 'Schulman-L-J', 'name': {'family': 'Schulman', 'given': 'Leonard J.'}, 'orcid': '0000-0001-9901-2797'}]}
Year: 1996
DOI: 10.1109/18.556671
Let the input to a computation problem be split between two processors connected by a communication link; and let an interactive protocol π be known by which, on any input, the processors can solve the problem using no more than T transmissions of bits between them, provided the channel is noiseless in each direction. We study the following question: if in fact the channel is noisy, what is the effect upon the number of transmissions needed in order to solve the computation problem reliably? Technologically this concern is motivated by the increasing importance of communication as a resource in computing, and by the tradeoff in communications equipment between bandwidth, reliability, and expense. We treat a model with random channel noise. We describe a deterministic method for simulating noiseless-channel protocols on noisy channels, with only a constant slowdown. This is an analog for general, interactive protocols of Shannon's coding theorem, which deals only with data transmission, i.e., one-way protocols. We cannot use Shannon's block coding method because the bits exchanged in the protocol are determined only one at a time, dynamically, in the course of the interaction. Instead, we describe a simulation protocol using a new kind of code, explicit tree codes.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/1gk21-mpy27The quantum communication complexity of sampling
https://resolver.caltech.edu/CaltechAUTHORS:20111213-142315946
Authors: {'items': [{'id': 'Ambainis-A', 'name': {'family': 'Ambainis', 'given': 'Andris'}}, {'id': 'Schulman-L-J', 'name': {'family': 'Schulman', 'given': 'Leonard J.'}, 'orcid': '0000-0001-9901-2797'}, {'id': 'Ta-Shma-A', 'name': {'family': 'Ta-Shma', 'given': 'Amnon'}}, {'id': 'Vazirani-Umesh-V', 'name': {'family': 'Vazirani', 'given': 'Umesh'}}, {'id': 'Wigderson-Avi', 'name': {'family': 'Wigderson', 'given': 'Avi'}}]}
Year: 1998
DOI: 10.1109/SFCS.1998.743480
Sampling is an important primitive in probabilistic and quantum algorithms. In the spirit of communication complexity, given a function f: X × Y → {0,1} and a probability distribution D over X × Y, we define the sampling complexity of (f,D) as the minimum number of bits Alice and Bob must communicate for Alice to pick x ∈ X and Bob to pick y ∈ Y as well as a valve z s.t. the resulting distribution of (x,y,z) is close to the distribution (D,f(D)). In this paper we initiate the study of sampling complexity, in both the classical and quantum model. We give several variants of the definition. We completely characterize some of these tasks, and give upper and lower bounds on others. In particular this allows us to establish an exponential gap between quantum and classical sampling complexity, for the set disjointness function. This is the first exponential gap for any task where the classical probabilistic algorithm is allowed to err.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/e8y67-3px55Pattern Matching for Spatial Point Sets
https://resolver.caltech.edu/CaltechAUTHORS:20120117-151248754
Authors: {'items': [{'id': 'Cardoze-D-E', 'name': {'family': 'Cardoze', 'given': 'David E.'}}, {'id': 'Schulman-L-J', 'name': {'family': 'Schulman', 'given': 'Leonard J.'}, 'orcid': '0000-0001-9901-2797'}]}
Year: 1998
DOI: 10.1109/SFCS.1998.743439
Two sets of points in d-dimensional space are given: a data set D consisting of N points, and a pattern set or probe P consisting of k points. We address the problem of determining whether there is a transformation, among a specified group of transformations of the space, carrying P into or near (meaning at a small directed Hausdorff distance of) D. The groups we consider are translations and rigid motions. Runtimes of approximately O(nlogn) and O(n^dlogn) respectively are obtained (letting n=max{N,k} and omitting the effects of several secondary parameters). For translations, a runtime of approximately O(n(ak+1)log^2n) is obtained for the case that a constant fraction α<1 of the points of the probe is allowed to fail to match.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/cpr56-npp98Signal propagation and noisy circuits
https://resolver.caltech.edu/CaltechAUTHORS:EVAieeetit99
Authors: {'items': [{'id': 'Evans-W-S', 'name': {'family': 'Evans', 'given': 'William S.'}}, {'id': 'Schulman-L-J', 'name': {'family': 'Schulman', 'given': 'Leonard J.'}, 'orcid': '0000-0001-9901-2797'}]}
Year: 1999
DOI: 10.1109/18.796377
The information carried by a signal decays when the signal is corrupted by random noise. This occurs when a message is transmitted over a noisy channel, as well as when a noisy component performs computation. We first study this signal decay in the context of communication and obtain a tight bound on the rate at which information decreases as a signal crosses a noisy channel. We then use this information theoretic result to obtain depth lower bounds in the noisy circuit model of computation defined by von Neumann. In this model, each component fails (produces 1 instead of 0 or vice-versa) independently with a fixed probability, and yet the output of the circuit is required to be correct with high probability. Von Neumann showed how to construct circuits in this model that reliably compute a function and are no more than a constant factor deeper than noiseless circuits for the function. We provide a lower bound on the multiplicative increase in circuit depth necessary for reliable computation, and an upper bound on the maximum level of noise at which reliable computation is possible.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/mcye7-j6k13Asymptotically good codes correcting insertions, deletions, and transpositions
https://resolver.caltech.edu/CaltechAUTHORS:SCHUieeetit99
Authors: {'items': [{'id': 'Schulman-L-J', 'name': {'family': 'Schulman', 'given': 'Leonard J.'}, 'orcid': '0000-0001-9901-2797'}, {'id': 'Zuckerman-D', 'name': {'family': 'Zuckerman', 'given': 'David'}}]}
Year: 1999
DOI: 10.1109/18.796406
We present simple, polynomial time encodable and decodable codes which are asymptotically good for channels allowing insertions, deletions, and transpositions. As a corollary, they achieve exponential error probability in a stochastic model of insertion-deletion.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/0h454-kks87Quantum mechanical algorithms for the nonabelian hidden subgroup problem
https://resolver.caltech.edu/CaltechAUTHORS:20161031-162455553
Authors: {'items': [{'id': 'Grigni-M', 'name': {'family': 'Grigni', 'given': 'Michaelangelo'}}, {'id': 'Schulman-L-J', 'name': {'family': 'Schulman', 'given': 'Leonard J.'}, 'orcid': '0000-0001-9901-2797'}, {'id': 'Vazirani-Monica-J', 'name': {'family': 'Vazirani', 'given': 'Monica'}}, {'id': 'Vazirani-Umesh-V', 'name': {'family': 'Vazirani', 'given': 'Umesh'}}]}
Year: 2001
DOI: 10.1145/380752.380769
We provide positive and negative results concerning the "standard method" of identifying a hidden subgroup of a nonabelian group using a quantum computer.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/c0qkx-v8885Microcellular Systems, Random Walks, and Wave Propagation
https://resolver.caltech.edu/CaltechPARADISE:2002.ETR045
Authors: {'items': [{'id': 'Franceschetti-M', 'name': {'family': 'Franceschetti', 'given': 'Massimo'}}, {'id': 'Bruck-J', 'name': {'family': 'Bruck', 'given': 'Jehoshua'}, 'orcid': '0000-0001-8474-0812'}, {'id': 'Schulman-L-J', 'name': {'family': 'Schulman', 'given': 'Leonard J.'}, 'orcid': '0000-0001-9901-2797'}]}
Year: 2002
As the number of users of wireless services increases, the concept of using smaller
cell sizes becomes especially attractive because of its potential for capacity increase.
Current technology allows to build base stations for small cells in a cost effective
way, and telecommunication companies have started exploiting the new microcellular
concept in providing coverage to densely populated areas. Prediction of propagation
characteristics in this new scenario is essential for accurate link budget calculations in
network planning.
In this paper a new, simple model of wave propagation for microcellular systems
is applied to predict the path loss of a wireless channel. The model does not rely on
the classical theory of electromagnetic wave propagation, but it is entirely based on
probability theory. We consider the canonical scenario of a random environment of
partially absorbing scatterers and model the trajectory of each photon in the system
as a random walk. This model leads to a path loss formula that rather accurately (in comparison to other models and experimental data) describes the smooth transition
of power attenuation from an inverse square law with the distance to the transmitter
to an exponential attenuation as this distance is increased. This result can justify
empirical formulas that are often used for path loss prediction, characterized by a
breakpoint distance at which the exponent of a power law is increased from a value of
approximately 2 to a value in the range of 4 to 10.
Theoretical predictions of the model are validated by showing agreement with experimental data collected in the city of Rome, Italy.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/1ryzf-n1606Lower Bounds for Linear Locally Decodable Codes and Private Information Retrieval
https://resolver.caltech.edu/CaltechAUTHORS:20111102-143056331
Authors: {'items': [{'id': 'Goldreich-O', 'name': {'family': 'Goldreich', 'given': 'Oded'}}, {'id': 'Karloff-H', 'name': {'family': 'Karloff', 'given': 'Howard'}}, {'id': 'Schulman-L-J', 'name': {'family': 'Schulman', 'given': 'Leonard J.'}, 'orcid': '0000-0001-9901-2797'}, {'id': 'Trevisan-L', 'name': {'family': 'Trevisan', 'given': 'Luca'}}]}
Year: 2002
DOI: 10.1109/CCC.2002.1004353
We prove that if a linear error-correcting code C: {0, 1}^n → {0, 1}^m is such that a bit of the message can be probabilistically reconstructed by looking at two entries of a corrupted codeword, then m = 2^(Ω(n)). We also present several extensions of this result. We show a reduction from the complexity, of one-round, information-theoretic private information retrieval systems (with two servers) to locally decodable codes, and conclude that if all the servers' answers are linear combinations of the database content, then t = Ω(n/2^a), where t is the length of the user's query and a is the length of the servers' answers. Actually, 2^a can be replaced by O(a^k), where k is the number of bit locations in the answer that are actually inspected in the reconstruction.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/dpvb8-wyd20A random stacking process
https://resolver.caltech.edu/CaltechAUTHORS:20170409-075810379
Authors: {'items': [{'id': 'Schulman-L-J', 'name': {'family': 'Schulman', 'given': 'Leonard J.'}, 'orcid': '0000-0001-9901-2797'}]}
Year: 2002
DOI: 10.1016/S0012-365X(02)00512-5
We examine a generalization of one-dimensional random walks with one reflecting and one absorbing boundary, in which a range of strategies is available, and the goal is to avoid absorption for as long as possible; we show a sharp threshold for the walk parameter separating linear and exponential expected walk times.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/cg30s-mkc28The quantum communication complexity of sampling
https://resolver.caltech.edu/CaltechAUTHORS:AMBsiamjc03
Authors: {'items': [{'id': 'Ambainis-A', 'name': {'family': 'Ambainis', 'given': 'Andris'}}, {'id': 'Schulman-L-J', 'name': {'family': 'Schulman', 'given': 'Leonard J.'}, 'orcid': '0000-0001-9901-2797'}, {'id': 'Ta-Shma-A', 'name': {'family': 'Ta-Shma', 'given': 'Amnon'}}, {'id': 'Vazirani-Umesh-V', 'name': {'family': 'Vazirani', 'given': 'Umesh'}}, {'id': 'Wigderson-A', 'name': {'family': 'Wigderson', 'given': 'Avi'}}]}
Year: 2003
DOI: 10.1137/S009753979935476
Sampling is an important primitive in probabilistic and quantum algorithms. In the spirit of communication complexity, given a function f : X × Y → {0, 1} and a probability distribution D over X × Y , we define the sampling complexity of (f,D) as the minimum number of bits that Alice and Bob must communicate for Alice to pick x ∈ X and Bob to pick y ∈ Y as well as a value z such that the resulting distribution of (x, y, z) is close to the distribution (D, f(D)).
In this paper we initiate the study of sampling complexity, in both the classical and quantum models. We give several variants of a definition. We completely characterize some of these variants and give upper and lower bounds on others. In particular, this allows us to establish an exponential gap between quantum and classical sampling complexity for the set-disjointness function.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/hgcq6-gjx82On the maximum tolerable noise of k-input gates for reliable computation by formulas
https://resolver.caltech.edu/CaltechAUTHORS:EVAieeetit03
Authors: {'items': [{'id': 'Evans-W-S', 'name': {'family': 'Evans', 'given': 'William S.'}}, {'id': 'Schulman-L-J', 'name': {'family': 'Schulman', 'given': 'Leonard J.'}, 'orcid': '0000-0001-9901-2797'}]}
Year: 2003
DOI: 10.1109/TIT.2003.818405
We determine the precise threshold of component noise below which formulas composed of odd degree components can reliably compute all Boolean functions.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/ewv4p-bs704Improved Expansion of Random Cayley Graphs
https://resolver.caltech.edu/CaltechAUTHORS:20200518-134456297
Authors: {'items': [{'id': 'Loh-Po-Shen', 'name': {'family': 'Loh', 'given': 'Po-Shen'}}, {'id': 'Schulman-L-J', 'name': {'family': 'Schulman', 'given': 'Leonard J.'}, 'orcid': '0000-0001-9901-2797'}]}
Year: 2004
Alon and Roichman (1994) proved that for every ε > 0 there is a finite c(e) such that for any sufficiently large group
G, the expected value of the second largest (in absolute value) eigenvalue of the normalized adjacency matrix of
the Cayley graph with respect to c(ε) log |G| random elements is less than ε. We reduce the number of elements
to c(ε) logD(G) (for the same c), where D(G) is the sum of the dimensions of the irreducible representations of
G. In sufficiently non-abelian families of groups (as measured by these dimensions), logD(G) is asymptotically
(1/2) log |G|. As is well known, a small eigenvalue implies large graph expansion (and conversely); see Tanner
(1984) and Alon and Milman (1984, 1985). For any specified eigenvalue or expansion, therefore, random Cayley
graphs (of sufficiently non-abelian groups) require only half as many edges as was previously known.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/5sxrn-qcg43The power of basis selection in fourier sampling: hidden subgroup problems in affine groups
https://resolver.caltech.edu/CaltechAUTHORS:20161130-165912905
Authors: {'items': [{'id': 'Moore-C', 'name': {'family': 'Moore', 'given': 'Cristopher'}}, {'id': 'Rockmore-D', 'name': {'family': 'Rockmore', 'given': 'Daniel'}}, {'id': 'Russell-A', 'name': {'family': 'Russell', 'given': 'Alexander'}}, {'id': 'Schulman-L-J', 'name': {'family': 'Schulman', 'given': 'Leonard J.'}, 'orcid': '0000-0001-9901-2797'}]}
Year: 2004
Many quantum algorithms, including Shor's celebrated factoring and discrete log algorithms, proceed by reduction to a hidden subgroup problem, in which a unknown subgroup H of a group G must be determined from a quantum state ψ over G that is uniformly supported on a left coset of H. These hidden subgroup problems are typically solved by Fourier sampling: the quantum Fourier transform of ψ is computed and measured. When the underlying group is nonabelian, two important variants of the Fourier sampling paradigm have been identified: the weak standard method, where only representation names are measured, and the strong standard method, where full measurement (i.e., the row and column of the representation as well as its name) occurs. It has remained open whether the strong method is indeed stronger, that is, whether there are hidden subgroups that can be reconstructed via the strong method but not by the weak, or any other known, method.In this article, we settle this question in the affirmative. We show that hidden subgroups of semidirect products of the form ℤq × ℤp, where q | (p - 1) and q = p/polylog(p), can be efficiently determined by the strong standard method. Furthermore, the weak standard method and the "forgetful" abelian method are insufficient for these groups so that, in fact, it appears that use of the corresponding nonabelian representation theory is crucial. We extend this to an informationtheoretic solution for the hidden subgroup problem over the groups ℤq × ℤp where q | (p - 1) and, in particular, the affine groups Ap. Finally, we prove a simple closure property for the class of groups over which the hidden subgroup problem can be solved efficiently.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/kcn13-8m180Quantum Mechanical Algorithms for the Nonabelian Hidden Subgroup Problem
https://resolver.caltech.edu/CaltechAUTHORS:20190829-131534342
Authors: {'items': [{'id': 'Grigni-M', 'name': {'family': 'Grigni', 'given': 'Michelangelo'}}, {'id': 'Schulman-L-J', 'name': {'family': 'Schulman', 'given': 'Leonard J.'}, 'orcid': '0000-0001-9901-2797'}, {'id': 'Vazirani-Monica-J', 'name': {'family': 'Vazirani', 'given': 'Monica'}}, {'id': 'Vazirani-Umesh-V', 'name': {'family': 'Vazirani', 'given': 'Umesh'}}]}
Year: 2004
DOI: 10.1007/s00493-004-0009-8
We provide positive and negative results concerning the "standard method" of identifying a hidden subgroup of a nonabelian group using a quantum computer.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/np257-djj69Fair and efficient router congestion control
https://resolver.caltech.edu/CaltechAUTHORS:20161025-172050851
Authors: {'items': [{'id': 'Gao-Xiaojie', 'name': {'family': 'Gao', 'given': 'Xiaojie'}}, {'id': 'Jain-Kamal', 'name': {'family': 'Jain', 'given': 'Kamal'}}, {'id': 'Schulman-L-J', 'name': {'family': 'Schulman', 'given': 'Leonard J.'}, 'orcid': '0000-0001-9901-2797'}]}
Year: 2004
Congestion is a natural phenomenon in any network queuing system, and is unavoidable if the queuing system is operated near capacity. In this paper we study how to set the rules of a queuing system so that all the users have a self-interest in controlling congestion when it happens.
Routers in the internet respond to local congestion by dropping packets. But if packets are dropped indiscriminately, the effect can be to encourage senders to actually increase their transmission rates, worsening the congestion and destabilizing the system. Alternatively, and only slightly more preferably, the effect can be to arbitrarily let a few insistent senders take over most of the router capacity.
We approach this problem from first principles: a router packet-dropping protocol is a mechanism that sets up a game between the senders, who are in turn competing for link capacity. Our task is to design this mechanism so that the game equilibrium is desirable: high total rate is achieved and is shared widely among all senders. In addition, equilibrium should be reestablished quickly in response to changes in transmission rates. Our solution is based upon auction theory: in principle, although not always in practice, we drop packets of the highest-rate sender, in case of congestion. We will prove the game-theoretic merits of our method. We'll also describe a variant of the method with some further advantages that will be supported by network simulations.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/axmph-qvd87A random walk model of wave propagation
https://resolver.caltech.edu/CaltechAUTHORS:FRAieeetap04
Authors: {'items': [{'id': 'Franceschetti-M', 'name': {'family': 'Franceschetti', 'given': 'Massimo'}}, {'id': 'Bruck-J', 'name': {'family': 'Bruck', 'given': 'Jehoshua'}, 'orcid': '0000-0001-8474-0812'}, {'id': 'Schulman-L-J', 'name': {'family': 'Shulman', 'given': 'Leonard J.'}, 'orcid': '0000-0001-9901-2797'}]}
Year: 2004
DOI: 10.1109/TAP.2004.827540
This paper shows that a reasonably accurate description of propagation loss in small urban cells can be obtained with a simple stochastic model based on the theory of random walks, that accounts for only two parameters: the amount of clutter and the amount of absorption in the environment. Despite the simplifications of the model, the derived analytical solution correctly describes the smooth transition of power attenuation from an inverse square law with the distance to the transmitter, to an exponential attenuation as this distance is increased - as it is observed in practice. Our analysis suggests using a simple exponential path loss formula as an alternative to the empirical formulas that are often used for prediction. Results are validated by comparison with experimental data collected in a small urban cell.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/ekacz-8vk35Wave-packet scattering without kinematic entanglement: convergence of expectation values
https://resolver.caltech.edu/CaltechAUTHORS:SCHUieeetnano05
Authors: {'items': [{'id': 'Schulman-Lawrence-S', 'name': {'family': 'Schulman', 'given': 'Lawrence S.'}, 'orcid': '0000-0001-5715-0542'}, {'id': 'Schulman-L-J', 'name': {'family': 'Schulman', 'given': 'Leonard J.'}, 'orcid': '0000-0001-9901-2797'}]}
Year: 2005
DOI: 10.1109/TNANO.2004.840141
The wave packet spread of a particle in a collection of different mass particles, all with Gaussian wave functions, evolves to a value that is inversely proportional to the mass of the particle. The assumptions underlying this result and its derivation are reviewed. A mathematical demonstration of the convergence of an iteration central to this assertion is presented. Finally, the question of in-principle measurement of wave packet spread is taken up.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/bcrvh-peb49Rapid near-optimal VQ design with a deterministic data net
https://resolver.caltech.edu/CaltechAUTHORS:EFFisit04
Authors: {'items': [{'id': 'Effros-M', 'name': {'family': 'Effros', 'given': 'Michelle'}}, {'id': 'Schulman-L-J', 'name': {'family': 'Schulman', 'given': 'Leonard J.'}, 'orcid': '0000-0001-9901-2797'}]}
Year: 2005
DOI: 10.1109/ISIT.2004.1365336
We present a new algorithm for fixed-rate vector quantizer (VQ) design with deterministic data net. The algorithm also performs efficient VQ design for simply characterized continuous distributions. The algorithm also serves as an approximation algorithm for the d-dimensional fixed-rate operational distortion-rate function, extends to a variety of network VQ problems. The algorithm generalizes to give /spl epsiv/-approximation algorithms for many network VQ design problems. A few examples are multiresolution VQ (MRVQ), multiple description VQ (MDVQ), side information VQ (SIVQ), Broadcast VQ (BCVQ), joint source-channel VQ (JSCVQ) and remote source VQ (RSVQ).https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/29xf4-4th78Physical Limits of Heat-Bath Algorithmic Cooling
https://resolver.caltech.edu/CaltechAUTHORS:SCHUprl05
Authors: {'items': [{'id': 'Schulman-L-J', 'name': {'family': 'Schulman', 'given': 'Leonard J.'}, 'orcid': '0000-0001-9901-2797'}, {'id': 'Mor-T', 'name': {'family': 'Mor', 'given': 'Tal'}}, {'id': 'Weinstein-Y', 'name': {'family': 'Weinstein', 'given': 'Yossi'}}]}
Year: 2005
DOI: 10.1103/PhysRevLett.94.120501
Simultaneous near-certain preparation of qubits (quantum bits) in their ground states is a key hurdle in quantum computing proposals as varied as liquid-state NMR and ion traps. "Closed-system" cooling mechanisms are of limited applicability due to the need for a continual supply of ancillas for fault tolerance, and to the high initial temperatures of some systems. "Open-system" mechanisms are therefore required. We describe a new, efficient initialization procedure for such open systems. With this procedure, an n-qubit device that is originally maximally mixed, but is in contact with a heat bath of bias epsilon>>2-n, can be almost perfectly initialized. This performance is optimal due to a newly discovered threshold effect: for bias epsilon<<2-n no cooling procedure can, even in principle (running indefinitely without any decoherence), significantly initialize even a single qubit.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/7rbaq-q5m53Feedback control for router congestion resolution
https://resolver.caltech.edu/CaltechAUTHORS:20161025-172835471
Authors: {'items': [{'id': 'Gao-Xiaojie', 'name': {'family': 'Gao', 'given': 'Xiaojie'}}, {'id': 'Schulman-L-J', 'name': {'family': 'Schulman', 'given': 'Leonard J.'}, 'orcid': '0000-0001-9901-2797'}]}
Year: 2005
DOI: 10.1145/1073814.1073855
Queueing is a crucial component in effective router congestion control. If packets are dropped indiscriminately by the queueing system, in some cases, the effect can be to encourage senders to actually increase their transmission rates, worsening the congestion and destabilizing the system.
We approach this congestion problem from the point of view of the elementary concepts of game theory and control theory. We provide a queueing mechanism with feedback-control. Our analysis shows that the protocol achieves high throughput as well as fairness in allocating capacity among sources, while maintaining bounded queue lengths and responding dynamically to changes in network flow conditions. Perhaps most importantly, the new protocol is shown in network simulations to have superior ability (compared with previous solutions) to protect responsive flows (specifically TCP) against router flooding by multiple high-volume unresponsive (e.g., UDP) flows.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/4ngkk-44b97A Computationally Motivated Definition Of Parametric Estimation And Its Applications To The Gaussian Distribution
https://resolver.caltech.edu/CaltechAUTHORS:20190820-150854373
Authors: {'items': [{'id': 'Schulman-L-J', 'name': {'family': 'Schulman', 'given': 'Leonard J.'}, 'orcid': '0000-0001-9901-2797'}, {'id': 'Vazirani-V-V', 'name': {'family': 'Vazirani', 'given': 'Vijay V.'}}]}
Year: 2005
DOI: 10.1007/s00493-005-0028-4
We introduce a treatment of parametric estimation in which optimality of an estimator is measured in probability rather than in variance (the measure for which the strongest general results are known in statistics). Our motivation is that the quality of an approximation algorithm is measured by the probability that it fails to approximate the desired quantity within a set tolerance. We concentrate on the Gaussian distribution and show that the sample mean is the unique "best" estimator, in probability, for the mean of a Gaussian distribution. We also extend this method to general penalty functions and to multidimensional spherically symmetric Gaussians.
The algorithmic significance of studying the Gaussian distribution is established by showing that determining the average matching size in a graph is #P-hard, and moreover approximating it reduces to estimating the mean of a random variable that (under some mild conditions) has a distribution closely approximating a Gaussian. This random variable is (essentially) polynomial time samplable, thereby yielding an FPRAS for the problem.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/9fta2-dhr34Real-Time Coding for Multiple Access Channels
https://resolver.caltech.edu/CaltechAUTHORS:20110817-093537514
Authors: {'items': [{'id': 'Gao-X', 'name': {'family': 'Gao', 'given': 'Xiaojie'}}, {'id': 'Schulman-L-J', 'name': {'family': 'Schulman', 'given': 'Leonard J.'}, 'orcid': '0000-0001-9901-2797'}]}
Year: 2005
DOI: 10.1109/ISIT.2005.1523294
We consider a multiple access channel shared by two sources. The channel is noiseless but there is interference between the transmissions of the sources. Because of applications to distributed control we are interested in the real-time version of this problem, in which the receiver must act immediately upon received information. Block coding is therefore not possible, and error probability cannot generally be made to tend to 0 in the interior of the multiple access capacity region. We study code design for a simple class of XOR channels. We provide several computationally efficient design methods. Under an assumption on the form of the correlation among the sources, one of these algorithms provides codes whose success probability is within 2/3 of optimal. In the absence of assumptions on the correlation, optimal code design is NP-hard.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/hbz9e-ag880The symmetric group defies strong Fourier sampling
https://resolver.caltech.edu/CaltechAUTHORS:20110822-081948476
Authors: {'items': [{'id': 'Moore-C', 'name': {'family': 'Moore', 'given': 'Cristopher'}}, {'id': 'Russell-A', 'name': {'family': 'Russell', 'given': 'Alexander'}}, {'id': 'Schulman-L-J', 'name': {'family': 'Schulman', 'given': 'Leonard J.'}, 'orcid': '0000-0001-9901-2797'}]}
Year: 2005
DOI: 10.1109/SFCS.2005.73
We resolve the question of whether Fourier sampling
can efficiently solve the hidden subgroup problem in general
groups. Specifically, we show that the hidden subgroup
problem in the symmetric group cannot be efficiently solved
by strong Fourier sampling. Indeed we prove the stronger
statement that no measurement of a single coset state can
reveal more than an exponentially small amount of information about the identity of the hidden subgroup, in the special case relevant to the Graph Isomorphism problem.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/hjtfv-bjs31Error-correcting codes for automatic control
https://resolver.caltech.edu/CaltechAUTHORS:20110824-152034750
Authors: {'items': [{'id': 'Ostrovsky-R', 'name': {'family': 'Ostrovsky', 'given': 'Rafail'}}, {'id': 'Rabani-Y', 'name': {'family': 'Rabani', 'given': 'Yuval'}}, {'id': 'Schulman-L-J', 'name': {'family': 'Schulman', 'given': 'Leonard J.'}, 'orcid': '0000-0001-9901-2797'}]}
Year: 2005
DOI: 10.1109/SFCS.2005.33
In many control-theory applications one can classify all possible states of the device by an infinite state graph with polynomially-growing expansion. In order for a controller to control or estimate the state of such a device, it must receive reliable communications from its sensors; if there is channel noise, the encoding task is subject to a stringent real-time constraint. We show a constructive on-line error correcting code that works for this class of applications. Our code is computationally efficient and enables on-line estimation and control in the presence of channel noise. It establishes a constructive (and optimal-within-constants) analog, for control applications, of the Shannon coding theorem.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/84d3g-fe475On Scalar LQG Control with Communication Cost
https://resolver.caltech.edu/CaltechAUTHORS:20110818-154937885
Authors: {'items': [{'id': 'Ko-C-K', 'name': {'family': 'Ko', 'given': 'Chih-Kai'}}, {'id': 'Gao-X', 'name': {'family': 'Gao', 'given': 'Xiaojie'}}, {'id': 'Prajna-S', 'name': {'family': 'Prajna', 'given': 'Stephen'}}, {'id': 'Schulman-L-J', 'name': {'family': 'Schulman', 'given': 'Leonard J.'}, 'orcid': '0000-0001-9901-2797'}]}
Year: 2005
DOI: 10.1109/CDC.2005.1582588
We study the LQG control of scalar systems under communication constraints by naturally extending the LQG cost to include a quadratic penalty for communication. We show that the resulting optimization problem is quasiconvex in the communications parameter so that it can be solved in a computationally efficient manner.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/wk81n-rqg78Analysis of incomplete data and an intrinsic-dimension Helly theorem
https://resolver.caltech.edu/CaltechAUTHORS:20161025-171347433
Authors: {'items': [{'id': 'Gao-Jie', 'name': {'family': 'Gao', 'given': 'Jie'}}, {'id': 'Langberg-M', 'name': {'family': 'Langberg', 'given': 'Michael'}, 'orcid': '0000-0002-7470-0718'}, {'id': 'Schulman-L-J', 'name': {'family': 'Schulman', 'given': 'Leonard J.'}, 'orcid': '0000-0001-9901-2797'}]}
Year: 2006
The analysis of incomplete data is a long-standing challenge in practical statistics. When, as is typical, data objects are represented by points in ℝ^d, incomplete data objects correspond to affine subspaces (lines or Δ-flats). With this motivation we study the problem of finding the minimum intersection radius r(L) of a set of lines or Δ-flats L: the least r such that there is a ball of radius r intersecting every flat in L. Known algorithms for finding the minimum enclosing ball for a point set (or clustering by several balls) do not easily extend to higher-dimensional flats, primarily because "distances" between flats do not satisfy the triangle inequality. In this paper we show how to restore geometry (i.e., a substitute for the triangle inequality) to the problem, through a new analog of Helly's theorem. This "intrinsic-dimension" Helly theorem states: for any family L of Δ-dimensional convex sets in a Hilbert space, there exist Δ + 2 sets L' ⊆ L such that r(L) ≤ 2r(L'). Based upon this we present an algorithm that computes a (1 + ε)-core set L' ⊆ L,|L'| = O(Δ^4/ε^2), such that the ball centered at a point c with radius (1 + ε)r(L') intersects every element of L. The running time of the algorithm is O(nΔ+1dpoly(1/ε)). For the case of lines or line segments (Δ = 1), the (expected) running time of the algorithm can be improved to O(nd poly(1/ε)). We note that the size of the core set depends only on the dimension of the input objects and is independent of the input size n and the dimension d of the ambient space.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/4k7mq-2bj70Convergence of matrices under random conjugation: wave packet scattering without kinematic entanglement
https://resolver.caltech.edu/CaltechAUTHORS:SCHUjpa06
Authors: {'items': [{'id': 'Schulman-Lawrence-S', 'name': {'family': 'Schulman', 'given': 'Lawrence S.'}, 'orcid': '0000-0001-5715-0542'}, {'id': 'Schulman-L-J', 'name': {'family': 'Schulman', 'given': 'Leonard J.'}, 'orcid': '0000-0001-9901-2797'}]}
Year: 2006
DOI: 10.1088/0305-4470/39/7/015
In previous work, it was shown numerically that under successive scattering events, a collection of particles with Gaussian wavefunctions retains the Gaussian property, with the spread of the Gaussian ('Δx') tending to a value inversely proportional to the square root of each particle's mass. We prove this convergence in all dimensions ≥3.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/apqk7-7bp48Computing with Highly Mixed States
https://resolver.caltech.edu/CaltechAUTHORS:20160419-113716190
Authors: {'items': [{'id': 'Ambainis-A', 'name': {'family': 'Ambainis', 'given': 'Andris'}}, {'id': 'Schulman-L-J', 'name': {'family': 'Schulman', 'given': 'Leonard J.'}, 'orcid': '0000-0001-9901-2797'}, {'id': 'Vazirani-Umesh-V', 'name': {'family': 'Vazirani', 'given': 'Umesh'}}]}
Year: 2006
DOI: 10.1145/1147954.1147962
Device initialization is a difficult challenge in some proposed realizations of quantum
computers, and as such, must be treated as a computational resource. The degree of initialization can
be quantified by k, the number of clean qubits in the initial state of the register. In this article, we
show that unless m ∈ O(k + log n), oblivious (gate-by-gate) simulation of an ideal m-qubit quantum
circuit by an n-qubit circuit with k clean qubits is impossible. Effectively, this indicates that there
is no avoiding physical initialization of a quantity of qubits proportional to that required by the best
ideal quantum circuit.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/tq5r6-bhp68Imaging geometry through dynamics: the observable representation
https://resolver.caltech.edu/CaltechAUTHORS:20090917-133724756
Authors: {'items': [{'id': 'Gaveau-B', 'name': {'family': 'Gaveau', 'given': 'Bernard'}}, {'id': 'Schulman-Lawrence-S', 'name': {'family': 'Schulman', 'given': 'Lawrence S.'}, 'orcid': '0000-0001-5715-0542'}, {'id': 'Schulman-L-J', 'name': {'family': 'Schulman', 'given': 'Leonard J.'}, 'orcid': '0000-0001-9901-2797'}]}
Year: 2006
DOI: 10.1088/0305-4470/39/33/004
For many stochastic processes there is an underlying coordinate space, V, with the process moving from point to point in V or on variables (such as spin configurations) defined with respect to V. There is a matrix of transition probabilities (whether between points in V or between variables defined on V) and we focus on its 'slow' eigenvectors, those with eigenvalues closest to that of the stationary eigenvector. These eigenvectors are the 'observables', and can be used to recover geometrical features of V.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/qh2xd-5m710The Effectiveness of Lloyd-Type Methods for the k-Means Problem
https://resolver.caltech.edu/CaltechAUTHORS:20170511-131811663
Authors: {'items': [{'id': 'Ostrovsky-R', 'name': {'family': 'Ostrovsky', 'given': 'Rafail'}}, {'id': 'Rabani-Y', 'name': {'family': 'Rabani', 'given': 'Yuval'}}, {'id': 'Schulman-L-J', 'name': {'family': 'Schulman', 'given': 'Leonard J.'}, 'orcid': '0000-0001-9901-2797'}, {'id': 'Swamy-C', 'name': {'family': 'Swamy', 'given': 'Chaitanya'}}]}
Year: 2006
DOI: 10.1109/FOCS.2006.75
We investigate variants of Lloyd's heuristic for clustering high dimensional data in an attempt to explain its popularity (a half century after its introduction) among practitioners, and in order to suggest improvements in its application. We propose and justify a clusterability criterion for data sets. We present variants of Lloyd's heuristic that quickly lead to provably near-optimal clustering solutions when applied to well-clusterable instances. This is the first performance guarantee for a variant of Lloyd's heuristic. The provision of a guarantee on output quality does not come at the expense of speed: some of our algorithms are candidates for being faster in practice than currently used variants of Lloyd's method. In addition, our other algorithms are faster on well-clusterable instances than recently proposed approximation algorithms, while maintaining similar guarantees on clustering quality. Our main algorithmic contribution is a novel probabilistic seeding process for the starting configuration of a Lloyd-type iteration.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/h917f-mb527Lower bounds for linear locally decodable codes and private information retrieval
https://resolver.caltech.edu/CaltechAUTHORS:20200225-113212660
Authors: {'items': [{'id': 'Goldreich-O', 'name': {'family': 'Goldreich', 'given': 'Oded'}}, {'id': 'Karloff-H', 'name': {'family': 'Karloff', 'given': 'Howard'}}, {'id': 'Schulman-L-J', 'name': {'family': 'Schulman', 'given': 'Leonard J.'}, 'orcid': '0000-0001-9901-2797'}, {'id': 'Trevisan-L', 'name': {'family': 'Trevisan', 'given': 'Luca'}}]}
Year: 2006
DOI: 10.1007/s00037-006-0216-3
We prove that if a linear error-correcting code C:{0, 1}^n→{0, 1}^m is such that a bit of the message can be probabilistically reconstructed by looking at two entries of a corrupted codeword, then m = 2^(Ω (n)). We also present several extensions of this result.
We show a reduction from the complexity of one-round, information-theoretic Private Information Retrieval Systems (with two servers) to Locally Decodable Codes, and conclude that if all the servers' answers are linear combinations of the database content, then t = Ω (n/2^a), where t is the length of the user's query and a is the length of the servers' answers. Actually, 2^a can be replaced by O(a^k), where k is the number of bit locations in the answer that are actually inspected in the reconstruction.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/cfyj6-2k048A Probabilistic Analysis of EM for Mixtures of Separated, Spherical Gaussians
https://resolver.caltech.edu/CaltechAUTHORS:DASjmlr07
Authors: {'items': [{'id': 'Dasgupta-Sanjoy', 'name': {'family': 'Dasgupta', 'given': 'Sanjoy'}}, {'id': 'Schulman-L-J', 'name': {'family': 'Schulman', 'given': 'Leonard J.'}, 'orcid': '0000-0001-9901-2797'}]}
Year: 2007
We show that, given data from a mixture of k well-separated spherical Gaussians in ℜ^d, a simple two-round variant of EM will, with high probability, learn the parameters of the Gaussians to near-optimal precision, if the dimension is high (d >> ln k). We relate this to previous theoretical and empirical work on the EM algorithm.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/2y8e7-40y25Physical Limits of Heat-Bath Algorithmic Cooling
https://resolver.caltech.edu/CaltechAUTHORS:SCHUsiamjc07
Authors: {'items': [{'id': 'Schulman-L-J', 'name': {'family': 'Schulman', 'given': 'Leonard J.'}, 'orcid': '0000-0001-9901-2797'}, {'id': 'Mor-T', 'name': {'family': 'Mor', 'given': 'Tal'}}, {'id': 'Weinstein-Y', 'name': {'family': 'Weinstein', 'given': 'Yossi'}}]}
Year: 2007
DOI: 10.1137/050666023
Simultaneous near-certain preparation of qubits (quantum bits) in their ground states is a key hurdle in quantum computing proposals as varied as liquid-state NMR and ion traps. "Closed-system" cooling mechanisms are of limited applicability due to the need for a continual supply of ancillas for fault tolerance and to the high initial temperatures of some systems. "Open-system" mechanisms are therefore required. We describe a new, efficient initialization procedure for such open systems. With this procedure, an $n$-qubit device that is originally maximally mixed, but is in contact with a heat bath of bias $\varepsilon \gg 2^{-n}$, can be almost perfectly initialized. This performance is optimal due to a newly discovered threshold effect: For bias $\varepsilon \ll 2^{-n}$ no cooling procedure can, even in principle (running indefinitely without any decoherence), significantly initialize even a single qubit.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/rxzxw-p5h47The Power of Strong Fourier Sampling: Quantum Algorithms for Affine Groups and Hidden Shifts
https://resolver.caltech.edu/CaltechAUTHORS:MOOsiamjc07
Authors: {'items': [{'id': 'Moore-C', 'name': {'family': 'Moore', 'given': 'Cristopher'}}, {'id': 'Rockmore-D', 'name': {'family': 'Rockmore', 'given': 'Daniel'}}, {'id': 'Russell-A', 'name': {'family': 'Russell', 'given': 'Alexander'}}, {'id': 'Schulman-L-J', 'name': {'family': 'Schulman', 'given': 'Leonard J.'}, 'orcid': '0000-0001-9901-2797'}]}
Year: 2007
DOI: 10.1137/S0097539705447177
Many quantum algorithms, including Shor's celebrated factoring and discrete log algorithms, proceed by reduction to a hidden subgroup problem, in which an unknown subgroup $H$ of a group $G$ must be determined from a quantum state $\psi$ over $G$ that is uniformly supported on a left coset of $H$. These hidden subgroup problems are typically solved by Fourier sampling: the quantum Fourier transform of $\psi$ is computed and measured. When the underlying group is nonabelian, two important variants of the Fourier sampling paradigm have been identified: the weak standard method, where only representation names are measured, and the strong standard method, where full measurement (i.e., the row and column of the representation, in a suitably chosen basis, as well as its name) occurs. It has remained open whether the strong standard method is indeed stronger, that is, whether there are hidden subgroups that can be reconstructed via the strong method but not by the weak, or any other known, method. In this article, we settle this question in the affirmative. We show that hidden subgroups $H$ of the $q$-hedral groups, i.e., semidirect products ${\mathbb Z}_q \ltimes {\mathbb Z}_p$, where $q \mid (p-1)$, and in particular the affine groups $A_p$, can be information-theoretically reconstructed using the strong standard method. Moreover, if $|H| = p/ {\rm polylog}(p)$, these subgroups can be fully reconstructed with a polynomial amount of quantum and classical computation. We compare our algorithms to two weaker methods that have been discussed in the literature—the "forgetful" abelian method, and measurement in a random basis—and show that both of these are weaker than the strong standard method. Thus, at least for some families of groups, it is crucial to use the full power of representation theory and nonabelian Fourier analysis, namely, to measure the high-dimensional representations in an adapted basis that respects the group's subgroup structure. We apply our algorithm for the hidden subgroup problem to new families of cryptographically motivated hidden shift problems, generalizing the work of van Dam, Hallgren, and Ip on shifts of multiplicative characters. Finally, we close by proving a simple closure property for the class of groups over which the hidden subgroup problem can be solved efficiently.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/8gh3w-2b995Approximation algorithms for labeling hierarchical taxonomies
https://resolver.caltech.edu/CaltechAUTHORS:20161212-164128630
Authors: {'items': [{'id': 'Rabani-Y', 'name': {'family': 'Rabani', 'given': 'Yuval'}}, {'id': 'Schulman-L-J', 'name': {'family': 'Schulman', 'given': 'Leonard J.'}, 'orcid': '0000-0001-9901-2797'}, {'id': 'Swamy-C', 'name': {'family': 'Swamy', 'given': 'Chaitanya'}}]}
Year: 2008
We consider the following taxonomy labeling problem. Each node of an n-node tree has to be labeled with the values of k attributes. A partial labeling is given as part of the input. The goal is to complete this labeling, minimizing the maximum variation in labeling along an edge. A special case of this problem (which we call the label extension problem), where every node is either completely labeled or not labeled at all, has been considered previously.
We present an O(log^2 k)-approximation algorithm based on a natural linear programming relaxation. Our results reduce the taxonomy labeling problem to another problem we introduce, called the multicut packing problem (on trees): given k multicommodity flow instances, find a multicut for each instance so as to minimize the maximum number of multicuts that use any single edge. Our algorithm yields an O(log^2 k)-approximation algorithm for this more general problem. We show that the integrality gap of our relaxation is Ω(log k), even when applied to the taxonomy labeling problem with 0-1 labels.
For the label extension problem, we considerably improve the previous O(log n) approximation guarantee and give the first constant-factor approximation algorithm for this problem. Our work relies on relating the label extension problem to questions on Lipschitz extensions of functions into Banach spaces. In particular, our approximation algorithm builds upon Matoušek's tree metrics extension theorem. Our algorithm also works for other metrics on the label-set, such as edit distance with unit-cost operations, and more generally any shortest path metric induced by an unweighted graph.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/91kwf-y5f84The Symmetric Group Defies Strong Fourier Sampling
https://resolver.caltech.edu/CaltechAUTHORS:MOOsiamjc08
Authors: {'items': [{'id': 'Moore-C', 'name': {'family': 'Moore', 'given': 'Christopher'}}, {'id': 'Russell-A', 'name': {'family': 'Russell', 'given': 'Alexander'}}, {'id': 'Schulman-L-J', 'name': {'family': 'Schulman', 'given': 'Leonard J.'}, 'orcid': '0000-0001-9901-2797'}]}
Year: 2008
DOI: 10.1137/050644896
The dramatic exponential speedups of quantum algorithms over their best existing classical counterparts were ushered in by the technique of Fourier sampling, introduced by Bernstein and Vazirani and developed by Simon and Shor into an approach to the hidden subgroup problem. This approach has proved successful for abelian groups, leading to efficient algorithms for factoring, extracting discrete logarithms, and other number-theoretic problems. We show, however, that this method cannot resolve the hidden subgroup problem in the symmetric groups, even in the weakest, information-theoretic sense. In particular, we show that the Graph Isomorphism problem cannot be solved by this approach. Our work implies that any quantum approach based upon the measurement of coset states must depart from the original framework by using entangled measurements on multiple coset states.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/c3f6z-nhz24On partitioning graphs via single commodity flows
https://resolver.caltech.edu/CaltechAUTHORS:20161206-174428027
Authors: {'items': [{'id': 'Orecchia-Lorenzo', 'name': {'family': 'Orecchia', 'given': 'Lorenzo'}}, {'id': 'Schulman-L-J', 'name': {'family': 'Schulman', 'given': 'Leonard J.'}, 'orcid': '0000-0001-9901-2797'}, {'id': 'Vazirani-Umesh-V', 'name': {'family': 'Vazirani', 'given': 'Umesh V.'}}, {'id': 'Vishnoi-Nisheet-K', 'name': {'family': 'Vishnoi', 'given': 'Nisheet K.'}}]}
Year: 2008
DOI: 10.1145/1374376.1374442
In this paper we obtain improved upper and lower bounds for the best approximation factor for Sparsest Cut achievable in the cut-matching game framework proposed in Khandekar et al. [9]. We show that this simple framework can be used to design combinatorial algorithms that achieve O(log n) approximation factor and whose running time is dominated by a poly-logarithmic number of single-commodity max-flow computations. This matches the performance of the algorithm of Arora and Kale [2]. Moreover, we also show that it is impossible to get an approximation factor of better than Ω(√log n) in the cut-matching game framework. These results suggest that the simple and concrete abstraction of the cut-matching game may be powerful enough to capture the essential features of the complexity of Sparsest Cut.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/yr4jp-anj81On a capacitated multivehicle routing problem
https://resolver.caltech.edu/CaltechAUTHORS:20161025-173435709
Authors: {'items': [{'id': 'Gao-Xiaojie', 'name': {'family': 'Gao', 'given': 'Xiaojie'}}, {'id': 'Schulman-L-J', 'name': {'family': 'Schulman', 'given': 'Leonard J.'}, 'orcid': '0000-0001-9901-2797'}]}
Year: 2008
DOI: 10.1145/1400751.1400776
The Vehicle Routing Problem (VRP) is a discrete optimization problem with high industrial relevance and high computational complexity. The problem has been extensively studied since it was introduced by Dantzig and Ramser. In this paper, we present a version of the VRP motivated by mobile sensor networks which we call the Capacitated Multivehicle Routing Problem (CMVRP). Our objective is to determine the minimum amount of energy required to serve all jobs, which takes into account both the service requirement and the travel overhead. We present a constant factor approximation algorithm for the off-line case and a distributed algorithm for the on-line problem that uses only a constant factor more energy than the off-line solution.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/m2xhy-7zg24Analysis of Incomplete Data and an Intrinsic-Dimension Helly Theorem
https://resolver.caltech.edu/CaltechAUTHORS:GAOdcg08
Authors: {'items': [{'id': 'Gao-Jie', 'name': {'family': 'Gao', 'given': 'Jie'}}, {'id': 'Langberg-M', 'name': {'family': 'Langberg', 'given': 'Michael'}, 'orcid': '0000-0002-7470-0718'}, {'id': 'Schulman-L-J', 'name': {'family': 'Schulman', 'given': 'Leonard J.'}, 'orcid': '0000-0001-9901-2797'}]}
Year: 2008
DOI: 10.1007/s00454-008-9107-5
The analysis of incomplete data is a long-standing challenge in practical statistics. When, as is typical, data objects are represented by points in R^d , incomplete data objects correspond to affine subspaces (lines or Δ-flats).With this motivation we study the problem of finding the minimum intersection radius r(L) of a set of lines or Δ-flats L: the least r such that there is a ball of radius r intersecting every flat in L. Known algorithms for finding the minimum enclosing ball for a point set (or clustering by several balls) do not easily extend to higher dimensional flats, primarily because "distances" between flats do not satisfy the triangle inequality. In this paper we show how to restore geometry (i.e., a substitute for the triangle inequality) to the problem, through a new analog of Helly's theorem. This "intrinsic-dimension" Helly theorem states: for any family L of Δ-dimensional convex sets in a Hilbert space, there exist Δ + 2 sets L' ⊆ L such that r(L) ≤ 2r(L'). Based upon this we present
an algorithm that computes a (1+ε)-core set L' ⊆ L, |L'| = O(Δ^4/ε), such that the ball centered at a point c with radius (1 +ε)r(L') intersects every element of L. The running time of the algorithm is O(n^(Δ+1)dpoly(Δ/ε)). For the case of lines or line segments (Δ = 1), the (expected) running time of the algorithm can be improved to O(ndpoly(1/ε)).We note that the size of the core set depends only on the dimension of the input objects and is independent of the input size n and the dimension d of the ambient space.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/fefaj-w5117Universal immersion spaces for edge-colored graphs and nearest-neighbor metrics
https://resolver.caltech.edu/CaltechAUTHORS:20090730-142657667
Authors: {'items': [{'id': 'Bartal-Y', 'name': {'family': 'Bartal', 'given': 'Yair'}}, {'id': 'Schulman-L-J', 'name': {'family': 'Schulman', 'given': 'Leonard J.'}, 'orcid': '0000-0001-9901-2797'}]}
Year: 2009
DOI: 10.1137/08071555X
There exist finite universal immersion spaces for the following: (a) Edge-colored graphs of bounded degree and boundedly many colors. (b) Nearest-neighbor metrics of bounded degree and boundedly many edge lengths.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/0b4c4-4d047Error-Correcting Codes for Automatic Control
https://resolver.caltech.edu/CaltechAUTHORS:20090831-141451771
Authors: {'items': [{'id': 'Ostrovsky-R', 'name': {'family': 'Ostrovsky', 'given': 'Rafail'}}, {'id': 'Rabani-Y', 'name': {'family': 'Rabani', 'given': 'Yuval'}}, {'id': 'Schulman-L-J', 'name': {'family': 'Schulman', 'given': 'Leonard J.'}, 'orcid': '0000-0001-9901-2797'}]}
Year: 2009
DOI: 10.1109/TIT.2009.2021303
Systems with automatic feedback control may consist of several remote devices, connected only by unreliable communication channels. It is necessary in these conditions to have a method for accurate, real-time state estimation in the presence of channel noise. This problem is addressed, for the case of polynomial-growth-rate state spaces, through a new type of error-correcting code that is online and computationally efficient. This solution establishes a constructive analog, for some applications in estimation and control, of the Shannon coding theorem.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/90apn-2kp46Muirhead-Rado inequality for compact groups
https://resolver.caltech.edu/CaltechAUTHORS:20090824-094016673
Authors: {'items': [{'id': 'Schulman-L-J', 'name': {'family': 'Schulman', 'given': 'Leonard J.'}, 'orcid': '0000-0001-9901-2797'}]}
Year: 2009
DOI: 10.1007/s11117-008-2172-4
Muirhead's majorization inequality was extended by Rado to the case of arbitrary permutation groups. We further generalize this inequality to compact groups and their linear representations over the reals. We characterize saturation of the inequality, and describe the saturation condition in detail for the case of actions on Hermitian operators.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/qm8yr-myq88Contraction and Expansion of Convex Sets
https://resolver.caltech.edu/CaltechAUTHORS:20091130-102029397
Authors: {'items': [{'id': 'Langberg-M', 'name': {'family': 'Langberg', 'given': 'Michael'}, 'orcid': '0000-0002-7470-0718'}, {'id': 'Schulman-L-J', 'name': {'family': 'Schulman', 'given': 'Leonard J.'}, 'orcid': '0000-0001-9901-2797'}]}
Year: 2009
DOI: 10.1007/s00454-009-9214-y
Let S be a set system of convex sets in R^d . Helly's theorem states that if all sets in S have empty intersection, then there is a subset S' ⊂ S of size d+1 which also has empty intersection. The conclusion fails, of course, if the sets in S are not convex or if S does not have empty intersection. Nevertheless, in this work we present Helly-type theorems relevant to these cases with the aid of a new pair of operations, affine-invariant contraction, and expansion of convex sets.
These operations generalize the simple scaling of centrally symmetric sets. The operations are continuous, i.e., for small ε>0, the contraction C^(−ε) and the expansion C^ε are close (in the Hausdorff distance) to C. We obtain two results. The first extends Helly's theorem to the case of set systems with nonempty intersection:
(a) If S is any family of convex sets in R^d , then there is a finite subfamily S' ⊆ S whose cardinality depends only on ε and d, such that ⋂_(C∈S')C^(−ε)⊆⋂_(C∈S)C.
The second result allows the sets in S a limited type of nonconvexity:
(b) If S is a family of sets in R^d, each of which is the union of k fat convex sets, then there is a finite subfamily S' ⊆ S whose cardinality depends only on ε, d, and k, such that ⋂_(C∈S')C^(−ε)⊆⋂_(C∈S)C.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/msxxy-aek26Variation on a theorem by Carathéodory
https://resolver.caltech.edu/CaltechAUTHORS:20100628-100808011
Authors: {'items': [{'id': 'Schulman-L-J', 'name': {'family': 'Schulman', 'given': 'Leonard J.'}, 'orcid': '0000-0001-9901-2797'}]}
Year: 2010
DOI: 10.1112/S0025579309000515
Carathéodory's theorem on small witnesses for convex hulls of sets is shown to have a natural analogue for finitely supported measures. Contrast is drawn with the much larger witnesses required for multisets, as shown by Bárány and Perles.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/ypn80-6m171Universal ε-approximators for integrals
https://resolver.caltech.edu/CaltechAUTHORS:20161121-163001649
Authors: {'items': [{'id': 'Langberg-M', 'name': {'family': 'Langberg', 'given': 'Michael'}, 'orcid': '0000-0002-7470-0718'}, {'id': 'Schulman-L-J', 'name': {'family': 'Schulman', 'given': 'Leonard J.'}, 'orcid': '0000-0001-9901-2797'}]}
Year: 2010
DOI: 10.1137/1.9781611973075.50
Let X be a space and F a family of 0, 1-valued functions on X. Vapnik and Chervonenkis showed that if F is "simple" (finite VC dimension), then for every probability measure μ on X and ε > 0 there is a finite set S such that for all f ∊ F, σx∊S f(x)/|S| = [∫ f (x)dμ(x)] ± ε.
Think of S as a "universal ε-approximator" for integration in F. S can actually be obtained w.h.p. just by sampling a few points from μ. This is a mainstay of computational learning theory. It was later extended by other authors to families of bounded (e.g., [0, 1]-valued) real functions.
In this work we establish similar "universal ε-approximators" for families of unbounded nonnegative real functions — in particular, for the families over which one optimizes when performing data classification. (In this case the ε-approximation should be multiplicative.)
Specifically, let F be the family of "k-median functions" (or k-means, etc.) on ℝd with an arbitrary norm ϱ. That is, any set u1, …, uk ∊ ℝd determines an f by f(x) = (mini ϱ(x – ui))α. (Here α ≥ 0.) Then for every measure μ on ℝd there exists a set S of cardinality poly(k, d, 1/ε) and a measure ν supported on S such that for every f ∊ F, σx∊S f(x)v(x) ∊ (1 ± ε) · (∫ f(x)dμ(x))https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/9n833-z8151Clustering Lines in High-Dimensional Space: Classification
of Incomplete Data
https://resolver.caltech.edu/CaltechAUTHORS:20110421-134437164
Authors: {'items': [{'id': 'Gao-J', 'name': {'family': 'Gao', 'given': 'Jie'}}, {'id': 'Langberg-M', 'name': {'family': 'Langberg', 'given': 'Michael'}, 'orcid': '0000-0002-7470-0718'}, {'id': 'Schulman-L-J', 'name': {'family': 'Schulman', 'given': 'Leonard J.'}, 'orcid': '0000-0001-9901-2797'}]}
Year: 2010
DOI: 10.1145/1868237.1868246
A set of k balls B_1,...,B_k in a Euclidean space is said to cover a collection of lines if every line intersects some ball. We consider the k-center problem for lines in high-dimensional space: Given a set of n lines ^I= {I_1,...,l_n in R^d, find k balls of minimum radius which cover I. We present a 2-approximation algorithm for the cases k = 2, 3 of this problem, having running time quasi-linear in the number of lines and the dimension of the ambient space. Our result for 3-clustering is strongly based on a new result in discrete geometry that may be of independent interest: a Helly-type theorem for collections of axis-parallel "crosses" in the plane. The family of crosses does not have finite Helly number in the usual sense. Our Helly theorem is of a new type: it depends on ε-contracting the sets.
In statistical practice, data is often incompletely specified; we consider lines as the most elementary case of incompletely specified data points. Clustering of data is a key primitive in nonparametric statistics. Our results provide a way of performing this primitive on incomplete data, as well as imputing the missing values.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/yhwnt-q8v98Dimensionality reduction: beyond the Johnson-Lindenstrauss bound
https://resolver.caltech.edu/CaltechAUTHORS:20120509-070406524
Authors: {'items': [{'id': 'Bartal-Y', 'name': {'family': 'Bartal', 'given': 'Yair'}}, {'id': 'Recht-B', 'name': {'family': 'Recht', 'given': 'Ben'}}, {'id': 'Schulman-L-J', 'name': {'family': 'Schulman', 'given': 'Leonard J.'}, 'orcid': '0000-0001-9901-2797'}]}
Year: 2011
Dimension reduction of metric data has become a useful technique with numerous applications. The celebrated Johnson-Lindenstrauss lemma states that any n-point subset of Euclidean space can be embedded in O(ε^(−2)log n)-dimension with (1 + ε)-distortion. This bound is known to be nearly tight.
In many applications the demand that all distances should be nearly preserved is too strong. In this paper we show that indeed under natural relaxations of the goal of the embedding, an improved dimension reduction is possible where the target dimension is independent of n. Our main result can be viewed as a local dimension reduction. There are a variety of empirical situations in which small distances are meaningful and reliable, but larger ones are not. Such situations arise in source coding, image processing, computational biology, and other applications, and are the motivation for widely-used heuristics such as Isomap and Locally Linear Embedding.
Pursuing a line of work begun by Whitney, Nash showed that every C^1 manifold of dimension d can be embedded in R^(2d+2) in such a manner that the local structure at each point is preserved isometrically. Our work is an analog of Nash's for discrete subsets of Euclidean space. For perfect preservation of infinitesimal neighborhoods we substitute near-isometric embedding of neighborhoods of bounded cardinality. We show that any finite subset of Euclidean space can be embedded in O(ε^(−2)log k)-dimension while preserving with (1 + ε)-distortion the distances within a "core neighborhood" of each point. (The core neighborhood is a metric ball around the point, whose radius is a substantial fraction of the radius of the ball of cardinality k, the k-neighborhood.) When the metric space satisfies a weak growth rate property, the guarantee applies to the entire k-neighborhood (with some dependency of the embedding dimension on the growth rate). We also show how to obtain a global embedding that also keeps distant points well-separated (at the cost of dependency on the doubling dimension of the space).
As an application of our methods we obtain an (Assouad-style) dimension reduction for finite subsets of Euclidean space where the metric is raised to some fractional power (the resulting metrics are known as snowflakes). We show that any such metric X can be embedded in dimension Õ(ε^(−3) dim(X)) with 1 + ε distortion, where dim(X) is the doubling dimension, a measure of the intrinsic dimension of the set. This result improves recent work by Gottlieb and Krauthgamer [20] to a nearly tight bound.
The new dimension reduction results are useful for applications such as clustering and distance labeling.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/mmhdh-yav65The quantifier semigroup for bipartite graphs
https://resolver.caltech.edu/CaltechAUTHORS:20110620-085724958
Authors: {'items': [{'id': 'Schulman-L-J', 'name': {'family': 'Schulman', 'given': 'Leonard J.'}, 'orcid': '0000-0001-9901-2797'}]}
Year: 2011
DOI: 10.37236/610
In a bipartite graph there are two widely encountered monotone mappings from subsets of one side of the graph to subsets of the other side: one corresponds to the quantifier "there exists a neighbor in the subset" and the other to the quantifier "all neighbors are in the subset." These mappings generate a partially ordered semigroup which we characterize in terms of "run-unimodal" words.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/hw9y3-6q494Data reduction for weighted and outlier-resistant clustering
https://resolver.caltech.edu/CaltechAUTHORS:20120524-112316051
Authors: {'items': [{'id': 'Feldman-D', 'name': {'family': 'Feldman', 'given': 'Dan'}}, {'id': 'Schulman-L-J', 'name': {'family': 'Schulman', 'given': 'Leonard J.'}, 'orcid': '0000-0001-9901-2797'}]}
Year: 2012
Statistical data frequently includes outliers; these can distort the results of estimation procedures and optimization problems. For this reason, loss functions which deemphasize the effect of outliers are widely used by statisticians. However, there are relatively few algorithmic results about clustering with outliers.
For instance, the k-median with outliers problem uses a loss function fc_1,...,c_k(x) which is equal to the minimum of a penalty h, and the least distance between the data point x and a center c_i. The loss-minimizing choice of {c_1,..., c_k} is an outlier-resistant clustering of the data. This problem is also a natural special case of the k-median with penalties problem considered by [Charikar, Khuller, Mount and Narasimhan SODA'01].
The essential challenge that arises in these optimization problems is data reduction for the weighted k-median problem. We solve this problem, which was previously solved only in one dimension ([Har-Peled FSTTCS'06], [Feldman, Fiat and Sharir FOCS'06]). As a corollary, we also achieve improved data reduction for the k-line-median problem.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/8b2fj-66088Cryptography from tensor problems
https://resolver.caltech.edu/CaltechAUTHORS:20120713-075312396
Authors: {'items': [{'id': 'Schulman-L-J', 'name': {'family': 'Schulman', 'given': 'Leonard J.'}, 'orcid': '0000-0001-9901-2797'}]}
Year: 2012
We describe a new proposal for a trap-door one-way function. The new proposal belongs to the "multivariate quadratic" family but the trap-door is different from existing methods, and is simpler.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/8qd5x-9v245The Effectiveness of Lloyd-Type Methods for the k-Means Problem
https://resolver.caltech.edu/CaltechAUTHORS:20130125-143155195
Authors: {'items': [{'id': 'Ostrovsky-R', 'name': {'family': 'Ostrovsky', 'given': 'Rafail'}}, {'id': 'Rabani-Y', 'name': {'family': 'Rabani', 'given': 'Yuval'}}, {'id': 'Schulman-L-J', 'name': {'family': 'Schulman', 'given': 'Leonard J.'}, 'orcid': '0000-0001-9901-2797'}, {'id': 'Swamy-C', 'name': {'family': 'Swamy', 'given': 'Chaitanya'}}]}
Year: 2012
DOI: 10.1145/2395116.2395117
We investigate variants of Lloyd's heuristic for clustering high dimensional data in an attempt to explain its popularity (a half century after its introduction) among practitioners, and in order to suggest improvements in its application. We propose and justify a clusterability criterion for data sets. We present variants of Lloyd's heuristic that quickly lead to provably near-optimal clustering solutions when applied to well-clusterable instances. This is the first performance guarantee for a variant of Lloyd's heuristic. The provision of a guarantee on output quality does not come at the expense of speed: some of our algorithms are candidates for being faster in practice than currently used variants of Lloyd's method. In addition, our other algorithms are faster on well-clusterable instances than recently proposed approximation algorithms, while maintaining similar guarantees on clustering quality. Our main algorithmic contribution is a novel probabilistic seeding process for the starting configuration of a Lloyd-type iteration.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/tewn0-mep45Clustering affine subspaces: hardness and algorithms
https://resolver.caltech.edu/CaltechAUTHORS:20161121-172931655
Authors: {'items': [{'id': 'Lee-Euiwoong', 'name': {'family': 'Lee', 'given': 'Euiwoong'}}, {'id': 'Schulman-L-J', 'name': {'family': 'Schulman', 'given': 'Leonard J.'}, 'orcid': '0000-0001-9901-2797'}]}
Year: 2013
DOI: 10.1137/1.9781611973105.58
We study a generalization of the famous k-center problem where each object is an affine subspace of dimension Δ, and give either the first or significantly improved algorithms and hardness results for many combinations of parameters. This generalization from points (Δ = 0) is motivated by the analysis of incomplete data, a pervasive challenge in statistics: incomplete data objects in ℝd can be modeled as affine subspaces. We give three algorithmic results for different values of k, under the assumption that all subspaces are axis-parallel, the main case of interest because of the correspondence to missing entries in data tables.
1) k = 1: Two polynomial time approximation schemes which runs in poly (Δ, 1/∊)nd.
2) k = 2: O(Δ1/4)-approximation algorithm which runs in poly(n, d, Δ)
3) General k: Polynomial time approximation scheme which runs in
We also prove nearly matching hardness results; in both the general (not necessarily axis-parallel) case (for k ≥ 2) and in the axis-parallel case (for k ≥ 3), the running time of an approximation algorithm with any approximation ratio cannot be polynomial in even one of k and Δ, unless P = NP. Furthermore, assuming that the 3-SAT problem cannot be solved sub-exponentially, the dependence on both k and Δ must be exponential in the general case (in the axis-parallel case, only the dependence on k drops to . The simplicity of the first and the third algorithm suggests that they might be actually used in statistical applications. The second algorithm, which demonstrates a theoretical gap between the axis-parallel and general case for k = 2, displays a strong connection between geometric clustering and classical coloring problems on graphs and hypergraphs, via a new Helly-type theorem.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/33twp-b8r58An MQ/Code Cyptosystem Proposal
https://resolver.caltech.edu/CaltechAUTHORS:20140130-133600557
Authors: {'items': [{'id': 'Schulman-L-J', 'name': {'family': 'Schulman', 'given': 'Leonard J.'}, 'orcid': '0000-0001-9901-2797'}]}
Year: 2013
We describe a new trap-door (and PKC) proposal. The proposal is ``multivariate quadratic'' (relies on the hardness of solving systems of quadratic equations); it is also code-based, and uses the code-scrambling technique of McEliece (1978). However, in the new proposal, the error-correcting code is not revealed in the public key, which protects against the leading attacks on McEliece's method.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/kmjbn-35b51Optimal Coding for Streaming Authentication and Interactive Communication
https://resolver.caltech.edu/CaltechAUTHORS:20170427-170911967
Authors: {'items': [{'id': 'Franklin-M', 'name': {'family': 'Franklin', 'given': 'Matthew'}}, {'id': 'Gelles-R', 'name': {'family': 'Gelles', 'given': 'Ran'}}, {'id': 'Ostrovsky-R', 'name': {'family': 'Ostrovsky', 'given': 'Rafail'}}, {'id': 'Schulman-L-J', 'name': {'family': 'Schulman', 'given': 'Leonard J.'}, 'orcid': '0000-0001-9901-2797'}]}
Year: 2013
Error correction and message authentication are well studied in the literature, and various efficient solutions have been suggested and analyzed. This is however not the case for data streams in which
the message is very long, possibly infinite, and not known in advance to the sender.
Trivial solutions for error-correcting and authenticating data streams either suffer from a long delay at the receiver's end or cannot perform well when the communication channel is noisy.
In this work we suggest a constant-rate error-correction scheme and an efficient authentication scheme for data streams over a noisy channel (one-way communication, no feedback) in the shared-randomness model. Our first scheme does not assume shared randomness and
(non-efficiently) recovers a (1−2c)-fraction prefix of the stream sent so far, assuming the noise level is at most c12 . The length of the recovered prefix is tight.
To be able to overcome the c=12 barrier we relax the model and assume the parties pre-share a secret key. Under this assumption we show that for any given noise rate c1, there exists a scheme that correctly decodes a (1−c)-fraction of the stream sent so far with high probability, and moreover, the scheme is efficient.
Furthermore, if the noise rate exceeds c, the scheme aborts with high probability. We also show that no constant-rate authentication scheme recovers more than a (1−c)-fraction of the stream sent so far with non-negligible probability, thus the relation between the noise rate and recoverable fraction of the stream is tight, and our scheme is optimal.
Our techniques also apply to the task of interactive communication (two-way communication) over a noisy channel. In a recent paper, Braverman and Rao [STOC 2011] show that any function of two inputs has a constant-rate interactive protocol for two users that withstands a noise rate up to 1/4.
By assuming that the parties share a secret random string,
we extend this result and
construct an interactive protocol that
succeeds with overwhelming probability against
noise rates up to 1/2. We also show that no constant-rate protocol exists for noise rates above 1/2 for functions that require two-way communication. This is contrasted with our first result in which computing the "function" requires only one-way communication and the noise rate can go up to 1.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/tmydw-2gs51Optimal Coding for Streaming Authentication and Interactive Communication
https://resolver.caltech.edu/CaltechAUTHORS:20170427-165048383
Authors: {'items': [{'id': 'Franklin-M', 'name': {'family': 'Franklin', 'given': 'Matthew'}}, {'id': 'Gelles-R', 'name': {'family': 'Gelles', 'given': 'Ran'}}, {'id': 'Ostrovsky-R', 'name': {'family': 'Ostrovsky', 'given': 'Rafail'}}, {'id': 'Schulman-L-J', 'name': {'family': 'Schulman', 'given': 'Leonard J.'}, 'orcid': '0000-0001-9901-2797'}]}
Year: 2013
DOI: 10.1007/978-3-642-40084-1_15
Error correction and message authentication are well studied in the literature, and various efficient solutions have been suggested and analyzed. This is however not the case for data streams in which the message is very long, possibly infinite, and not known in advance to the sender. Trivial solutions for error-correcting and authenticating data streams either suffer from a long delay at the receiver's end or cannot perform well when the communication channel is noisy.
In this work we suggest a constant-rate error-correction scheme and an efficient authentication scheme for data streams over a noisy channel (one-way communication, no feedback) in the shared-randomness model. Our first scheme does not assume shared randomness and (non-efficiently) recovers a (1 − 2c)-fraction prefix of the stream sent so far, assuming the noise level is at most c < 1/2. The length of the recovered prefix is tight.
To be able to overcome the c = 1/2 barrier we relax the model and assume the parties pre-share a secret key. Under this assumption we show that for any given noise rate c < 1, there exists a scheme that correctly decodes a (1 − c)-fraction of the stream sent so far with high probability, and moreover, the scheme is efficient. Furthermore, if the noise rate exceeds c, the scheme aborts with high probability. We also show that no constant-rate authentication scheme recovers more than a (1 − c)-fraction of the stream sent so far with non-negligible probability, thus the relation between the noise rate and recoverable fraction of the stream is tight, and our scheme is optimal.
Our techniques also apply to the task of interactive communication (two-way communication) over a noisy channel. In a recent paper, Braverman and Rao [STOC 2011] show that any function of two inputs has a constant-rate interactive protocol for two users that withstands a noise rate up to 1/4. By assuming that the parties share a secret random string, we extend this result and construct an interactive protocol that succeeds with overwhelming probability against noise rates up to 1/2. We also show that no constant-rate protocol exists for noise rates above 1/2 for functions that require two-way communication. This is contrasted with our first result in which computing the "function" requires only one-way communication and the noise rate can go up to 1.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/kqzsq-8jg40Network Improvement for Equilibrium Routing
https://resolver.caltech.edu/CaltechAUTHORS:20150223-101614687
Authors: {'items': [{'id': 'Bhaskar-U', 'name': {'family': 'Bhaskar', 'given': 'Umang'}}, {'id': 'Ligett-K', 'name': {'family': 'Ligett', 'given': 'Katrina'}, 'orcid': '0000-0003-2780-6656'}, {'id': 'Schulman-L-J', 'name': {'family': 'Schulman', 'given': 'Leonard J.'}, 'orcid': '0000-0001-9901-2797'}]}
Year: 2014
In routing games, agents pick routes through a network to
minimize their own delay. A primary concern for the network designer
in routing games is the average agent delay at equilibrium. A number of
methods to control this average delay have received substantial attention,
including network tolls, Stackelberg routing, and edge removal.
A related approach with arguably greater practical relevance is that
of making investments in improvements to the edges of the network, so
that, for a given investment budget, the average delay at equilibrium in
the improved network is minimized. This problem has received considerable
attention in the literature on transportation research. We study
a model for this problem introduced in transportation research literature,
and present both hardness results and algorithms that obtain tight
performance guarantees.
– In general graphs, we show that a simple algorithm obtains a
4/3-approximation for affine delay functions and an O(p/ log p)-
approximation for polynomial delay functions of degree p. For affine
delays, we show that it is NP-hard to improve upon the 4/3 approximation.
– Motivated by the practical relevance of the problem, we consider restricted
topologies to obtain better bounds. In series-parallel graphs,
we show that the problem is still NP-hard. However, we show that
there is an FPTAS in this case.
– Finally, for graphs consisting of parallel paths, we show that an optimal
allocation can be obtained in polynomial time.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/59t3z-qw360Tree Codes and a Conjecture on Exponential Sums
https://resolver.caltech.edu/CaltechAUTHORS:20140331-152623727
Authors: {'items': [{'id': 'Moore-C', 'name': {'family': 'Moore', 'given': 'Cristopher'}}, {'id': 'Schulman-L-J', 'name': {'family': 'Schulman', 'given': 'Leonard J.'}, 'orcid': '0000-0001-9901-2797'}]}
Year: 2014
DOI: 10.1145/2554797.2554813
We propose a new conjecture on some exponential sums.
These particular sums have not apparently been considered
in the literature. Subject to the conjecture we obtain the
first effective construction of asymptotically good tree codes.
The available numerical evidence is consistent with the conjecture and is sufficient to certify codes for significant-length communications.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/vh65c-pd908Optimal Coding for Streaming Authentication and Interactive Communication
https://resolver.caltech.edu/CaltechAUTHORS:20150202-090030427
Authors: {'items': [{'id': 'Franklin-M', 'name': {'family': 'Franklin', 'given': 'Matthew'}}, {'id': 'Gelles-R', 'name': {'family': 'Gelles', 'given': 'Ran'}}, {'id': 'Ostrovsky-R', 'name': {'family': 'Ostrovsky', 'given': 'Rafail'}}, {'id': 'Schulman-L-J', 'name': {'family': 'Schulman', 'given': 'Leonard J.'}, 'orcid': '0000-0001-9901-2797'}]}
Year: 2014
DOI: 10.1109/TIT.2014.2367094
We consider the task of communicating a data stream-a long, possibly infinite message not known in advance to the sender-over a channel with adversarial noise. For any given noise rate c <; 1, we show an efficient, constant-rate scheme that correctly decodes a (1 - c) fraction of the stream sent so far with high probability, or aborts if the noise rate exceeds c. In addition, we prove that no constant-rate scheme can recover more than a (1 - c) fraction of the stream sent so far with non-negligible probability, which makes our scheme optimal in that aspect. The parties are assumed to preshare a random string unknown to the channel. Our techniques can also be applied to the task of interactive communication (two-way communication) over a noisy channel. In a recent paper (Braverman and Rao, STOC11), the possibility of two-party interactive communication as long as the noise level is <; 1/4 was shown. By allowing the parties to preshare some private random string, we extend this result and construct a (nonefficient) constant-rate interactive protocol that succeeds with overwhelming probability against noise rates up to 1/2. We complete this result by proving that no constant-rate protocol can withstand noise rates > 1/2.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/9zea3-qk087Learning mixtures of arbitrary distributions over large discrete domains
https://resolver.caltech.edu/CaltechAUTHORS:20161212-165501273
Authors: {'items': [{'id': 'Rabani-Y', 'name': {'family': 'Rabani', 'given': 'Yuval'}}, {'id': 'Schulman-L-J', 'name': {'family': 'Schulman', 'given': 'Leonard J.'}, 'orcid': '0000-0001-9901-2797'}, {'id': 'Swamy-C', 'name': {'family': 'Swamy', 'given': 'Chaitanya'}}]}
Year: 2014
DOI: 10.1145/2554797.2554818
We give an algorithm for learning a mixture of unstructured distributions. This problem arises in various unsupervised learning scenarios, for example in learning topic models from a corpus of documents spanning several topics. We show how to learn the constituents of a mixture of k arbitrary distributions over a large discrete domain [n]={1, 2, ...,n} and the mixture weights, using O(n polylog n) samples. (In the topic-model learning setting, the mixture constituents correspond to the topic distributions.)
This task is information-theoretically impossible for k > 1 under the usual sampling process from a mixture distribution. However, there are situations (such as the above-mentioned topic model case) in which each sample point consists of several observations from the same mixture constituent. This number of observations, which we call the "sampling aperture", is a crucial parameter of the problem.
We obtain the first bounds for this mixture-learning problem without imposing any assumptions on the mixture constituents. We show that efficient learning is possible exactly at the information-theoretically least-possible aperture of 2k-1. Thus, we achieve near-optimal dependence on n and optimal aperture. While the sample-size required by our algorithm depends exponentially on k, we prove that such a dependence is unavoidable when one considers general mixtures.
A sequence of tools contribute to the algorithm, such as concentration results for random matrices, dimension reduction, moment estimations, and sensitivity analysis.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/jtczr-4n629Dimension-free L_2 maximal inequality for spherical means in the hypercube
https://resolver.caltech.edu/CaltechAUTHORS:20130122-104220997
Authors: {'items': [{'id': 'Harrow-A-W', 'name': {'family': 'Harrow', 'given': 'Aram W.'}}, {'id': 'Kolla-A', 'name': {'family': 'Kolla', 'given': 'Alexandra'}}, {'id': 'Schulman-L-J', 'name': {'family': 'Schulman', 'given': 'Leonard J.'}, 'orcid': '0000-0001-9901-2797'}]}
Year: 2014
DOI: 10.4086/toc.2014.v010a003
We establish the maximal inequality claimed in the title. In combinatorial terms this has the implication that for sufficiently small ε > 0, for all n, any marking of an ε fraction of the vertices of the n-dimensional hypercube necessarily leaves a vertex x such that marked vertices are a minority of every sphere centered at x.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/2fvr9-gmf61Volume in General Metric Spaces
https://resolver.caltech.edu/CaltechAUTHORS:20141015-160530977
Authors: {'items': [{'id': 'Abraham-I', 'name': {'family': 'Abraham', 'given': 'Ittai'}}, {'id': 'Bartal-Y', 'name': {'family': 'Bartal', 'given': 'Yair'}}, {'id': 'Neiman-O', 'name': {'family': 'Neiman', 'given': 'Ofer'}}, {'id': 'Schulman-L-J', 'name': {'family': 'Schulman', 'given': 'Leonard J.'}, 'orcid': '0000-0001-9901-2797'}]}
Year: 2014
DOI: 10.1007/s00454-014-9615-4
A central question in the geometry of finite metric spaces is how well can an arbitrary metric space be "faithfully preserved" by a mapping into Euclidean space. In this paper we present an algorithmic embedding which obtains a new strong measure of faithful preservation: not only does it (approximately) preserve distances between pairs of points, but also the volume of any set of k points. Such embeddings are known as volume preserving embeddings. We provide the first volume preserving embedding that obtains constant average volume distortion for sets of any fixed size. Moreover, our embedding provides constant bounds on all bounded moments of the volume distortion while maintaining the best possible worst-case volume distortion. Feige, in his seminal work on volume preserving embeddings defined the volume of a set S={v_1,…,v_k} of points in a general metric space: the product of the distances from vi to {v_1,…,v_(i−1)}, normalized by 1/(k−1)!, where the ordering of the points is that given by Prim's minimum spanning tree algorithm. Feige also related this notion to the maximal Euclidean volume that a Lipschitz embedding of S into Euclidean space can achieve. Syntactically this definition is similar to the computation of volume in Euclidean spaces, which however is invariant to the order in which the points are taken. We show that a similar robustness property holds for Feige's definition: the use of any other order in the product affects volume ^(1/(k−1)) by only a constant factor. Our robustness result is of independent interest as it presents a new competitive analysis for the greedy algorithm on a variant of the online Steiner tree problem where the cost of buying an edge is logarithmic in its length. This robustness property allows us to obtain our results on volume preserving embedding.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/qahjw-gta16Achieving Target Equilibria in Network Routing Games without Knowing the Latency Functions
https://resolver.caltech.edu/CaltechAUTHORS:20160105-073143688
Authors: {'items': [{'id': 'Bhaskar-U', 'name': {'family': 'Bhaskar', 'given': 'Umang'}}, {'id': 'Ligett-K', 'name': {'family': 'Ligett', 'given': 'Katrina'}, 'orcid': '0000-0003-2780-6656'}, {'id': 'Schulman-L-J', 'name': {'family': 'Schulman', 'given': 'Leonard J.'}, 'orcid': '0000-0001-9901-2797'}, {'id': 'Swamy-Chaitanya', 'name': {'family': 'Swamy', 'given': 'Chaitanya'}}]}
Year: 2014
DOI: 10.1109/FOCS.2014.12
The analysis of network routing games typically assumes, right at the onset, precise and detailed information about the latency functions. Such information may, however, be unavailable or difficult to obtain. Moreover, one is often primarily interested in enforcing a desirable target flow as the equilibrium by suitably influencing player behavior in the routing game. We ask whether one can achieve target flows as equilibria without knowing the underlying latency functions. Our main result gives a crisp positive answer to this question. We show that, under fairly general settings, one can efficiently compute edge tolls that induce a given target multicommodity flow in a nonatomic routing game using a polynomial number of queries to an oracle that takes candidate tolls as input and returns the resulting equilibrium flow. This result is obtained via a novel application of the ellipsoid method, and applies to arbitrary multicommodity settings and non-linear latency functions. Our algorithm extends easily to many other settings, such as (i) when certain edges cannot be tolled or there is an upper bound on the total toll paid by a user, and (ii) general nonatomic congestion games. We obtain tighter bounds on the query complexity for series-parallel networks, and single-commodity routing games with linear latency functions, and complement these with a query-complexity lower bound applicable even to single-commodity routing games on parallel-link graphs with linear latency functions. We also explore the use of Stackelberg routing to achieve target equilibria and obtain strong positive results for series-parallel graphs. Our results build upon various new techniques that we develop pertaining to the computation of, and connections between, different notions of approximate equilibrium, properties of multicommodity flows and tolls in series-parallel graphs, and sensitivity of equilibrium flow with respect to tolls. Our results demonstrate that one can indeed circumvent the potentially-onerous task of modeling latency functions, and yet obtain meaningful results for the underlying routing game.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/4char-3ct96Learning Arbitrary Statistical Mixtures of Discrete Distributions
https://resolver.caltech.edu/CaltechAUTHORS:20150715-085732981
Authors: {'items': [{'id': 'Li-Jian', 'name': {'family': 'Li', 'given': 'Jian'}, 'orcid': '0000-0003-0297-6528'}, {'id': 'Rabani-Y', 'name': {'family': 'Rabani', 'given': 'Yuval'}}, {'id': 'Schulman-L-J', 'name': {'family': 'Schulman', 'given': 'Leonard J.'}, 'orcid': '0000-0001-9901-2797'}, {'id': 'Swamy-C', 'name': {'family': 'Swamy', 'given': 'Chaitanya'}}]}
Year: 2015
DOI: 10.1145/2746539.2746584
We study the problem of learning from unlabeled samples very general statistical mixture models on large finite sets. Specifically, the model to be learned, mix, is a probability distribution over probability distributions p, where each such p is a probability distribution over [n] = {1,2,...,n}. When we sample from mix, we do not observe p directly, but only indirectly and in very noisy fashion, by sampling from [n] repeatedly, independently K times from the distribution p. The problem is to infer mix to high accuracy in transportation (earthmover) distance.
We give the first efficient algorithms for learning this mixture model without making any restricting assumptions on the structure of the distribution mix. We bound the quality of the solution as a function of the size of the samples K and the number of samples used. Our model and results have applications to a variety of unsupervised learning scenarios, including learning topic models and collaborative filtering.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/97whm-jn174Analysis of a Classical Matrix Preconditioning Algorithm
https://resolver.caltech.edu/CaltechAUTHORS:20150715-085040447
Authors: {'items': [{'id': 'Schulman-L-J', 'name': {'family': 'Schulman', 'given': 'Leonard J.'}, 'orcid': '0000-0001-9901-2797'}, {'id': 'Sinclair-A', 'name': {'family': 'Sinclair', 'given': 'Alistair'}}]}
Year: 2015
DOI: 10.1145/2746539.2746556
We study a classical iterative algorithm for the problem of balancing matrices in the L∞ norm via a scaling transformation. This algorithm, which goes back to Osborne and Parlett & Reinsch in the 1960s, is implemented as a standard preconditioner in many numerical linear algebra packages. Surprisingly, despite its widespread use over several decades, no bounds were known on its rate of convergence. In this paper we prove that, for a large class of irreducible n x n (real or complex) input matrices A, a natural variant of the algorithm converges in O(n^3 log(nρ/ε)) elementary balancing operations, where ρ measures the initial imbalance of A and ε is the target imbalance of the output matrix. (The imbalance of A is maxi |log(a_i^(out)/a_i^(in))|, where a_i^(out),a_i^(in) are the maximum entries in magnitude in the ith row and column respectively.) This bound is tight up to the log n factor. A balancing operation scales the ith row and column so that their maximum entries are equal, and requires O(m/n) arithmetic operations on average, where m is the number of non-zero elements in A. Thus the running time of the iterative algorithm is ~O(n^2m). This is the first time bound of any kind on any variant of the Osborne-Parlett-Reinsch algorithm. The class of matrices for which the above analysis holds are those which satisfy a condition we call Unique Balance, meaning that the limit of the iterative balancing process does not depend on the order in which balancing operations are performed. We also prove a combinatorial characterization of the Unique Balance property, which had earlier been conjectured by Chen.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/45nqa-2ev81Symbolic Integration and the Complexity of Computing Averages
https://resolver.caltech.edu/CaltechAUTHORS:20160804-113612809
Authors: {'items': [{'id': 'Schulman-L-J', 'name': {'family': 'Schulman', 'given': 'Leonard J.'}, 'orcid': '0000-0001-9901-2797'}, {'id': 'Sinclair-A', 'name': {'family': 'Sinclair', 'given': 'Alistair'}}, {'id': 'Srivastava-P', 'name': {'family': 'Srivastava', 'given': 'Piyush'}, 'orcid': '0000-0003-0953-2890'}]}
Year: 2015
DOI: 10.1109/FOCS.2015.79
We study the computational complexity of several natural problems arising in statistical physics and combinatorics. In particular, we consider the following problems: the mean magnetization and mean energy of the Ising model (both the ferromagnetic and the anti-ferromagnetic settings), the average size of an independent set in the hard core model, and the average size of a matching in the monomer-dimer model. We prove that for all non-trivial values of the underlying model parameters, exactly computing these averages is #P-hard.
In contrast to previous results of Sinclair and Srivastava [1] for the mean magnetization of the ferromagnetic Ising model, our approach does not use any Lee-Yang type theorems about the complex zeros of partition functions. Indeed, it was due to the lack of suitable Lee-Yang theorems for models such as the anti-ferromagnetic Ising model that some of the problems we study here were left open in [1]. In this paper, we instead use some relatively simple and well-known ideas from the theory of automatic symbolic integration to complete our hardness reductions.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/js85e-0ah54The adversarial noise threshold for distributed protocols
https://resolver.caltech.edu/CaltechAUTHORS:20160316-071949447
Authors: {'items': [{'id': 'Hoza-W-M', 'name': {'family': 'Hoza', 'given': 'William M.'}}, {'id': 'Schulman-L-J', 'name': {'family': 'Schulman', 'given': 'Leonard J.'}, 'orcid': '0000-0001-9901-2797'}]}
Year: 2016
DOI: 10.48550/arXiv.1412.8097
We consider the problem of implementing distributed protocols, despite adversarial channel errors, on synchronous-messaging networks with arbitrary topology.
In our first result we show that any n-party T-round protocol on an undirected communication network G can be compiled into a robust simulation protocol on a sparse (O(n) edges) subnetwork so that the simulation tolerates an adversarial error rate of Ω(1n); the simulation has a round complexity of O(m log n/nT), where m is the number of edges in G. (So the simulation is work-preserving up to a log factor.) The adversary's error rate is within a constant factor of optimal. Given the error rate, the round complexity blowup is within a factor of O(k log n) of optimal, where k is the edge connectivity of G. We also determine that the maximum tolerable error rate on directed communication networks is Θ(1/s) where s is the number of edges in a minimum equivalent digraph.
Next we investigate adversarial per-edge error rates, where the adversary is given an error budget on each edge of the network. We determine the exact limit for tolerable per-edge error rates on an arbitrary directed graph. However, the construction that approaches this limit has exponential round complexity, so we give another compiler, which transforms T-round protocols into O(mT)-round simulations, and prove that for polynomial-query black box compilers, the per-edge error rate tolerated by this last compiler is within a constant factor of optimal.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/77mce-h7v87Stability of Causal Inference
https://resolver.caltech.edu/CaltechAUTHORS:20160628-152001110
Authors: {'items': [{'id': 'Schulman-L-J', 'name': {'family': 'Schulman', 'given': 'Leonard J.'}, 'orcid': '0000-0001-9901-2797'}, {'id': 'Srivastava-P', 'name': {'family': 'Srivastava', 'given': 'Piyush'}, 'orcid': '0000-0003-0953-2890'}]}
Year: 2016
We consider the sensitivity of causal identification
to small perturbations in the input. A long line of
work culminating in papers by Shpitser and Pearl
(2006) and Huang and Valtorta (2008) led to a
complete procedure for the causal identification
problem. In our main result in this paper, we
show that the identification function computed
by these procedures is in some cases extremely
unstable numerically. Specifically, the "condition
number" of causal identification can be of the
order of Ω(exp(n
^(0.49))) on an identifiable semiMarkovian
model with n visible nodes. That is,
in order to give an output accurate to d bits, the
empirical probabilities of the observable events
need to be obtained to accuracy d + Ω(n
^(0.49)) bits.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/n90z2-k3643Extractors for Near Logarithmic Min-Entropy
https://resolver.caltech.edu/CaltechAUTHORS:20170127-140321585
Authors: {'items': [{'id': 'Cohen-Gil', 'name': {'family': 'Cohen', 'given': 'Gil'}}, {'id': 'Schulman-L-J', 'name': {'family': 'Schulman', 'given': 'Leonard J.'}, 'orcid': '0000-0001-9901-2797'}]}
Year: 2016
DOI: 10.1109/FOCS.2016.27
The main contribution of this work is an explicit construction of extractors for near logarithmic min-entropy. For any δ > 0 we construct an extractor for O(1/δ) n-bit sources with min-entropy (logn)^(1+δ). This is most interesting when δ is set to a small constant, though the result also yields an extractor for O(log logn) sources with logarithmic min-entropy. Prior to this work, the best explicit extractor in terms of supporting least-possible min-entropy, due to Li (FOCS'15), requires min-entropy (logn)^(2+δ) from its O(1/δ) sources. Further, all current techniques for constructing multi-source extractors "break" below min-entropy (log n)^2. In fact, existing techniques do not provide even a disperser for o(log n) sources each with min-entropy (log n)^(1.99). Apart from being a natural problem, supporting logarithmic min-entropy has applications to combinatorics. A two-source disperser, let alone an extractor, for min-entropy O(log n) induces a (log, nO(1))-Ramsey graph on n vertices. Thus, constructing such dispersers would be a significant step towards constructively matching Erdös' proof for the existence of (2log n)-Ramsey graphs on n vertices. Our construction does not rely on the sophisticated primitives that were key to the substantial recent progress on multi-source extractors, such as non-malleable extractors, correlation breakers, the lightest-bin condenser, or extractors for non-oblivious bit-fixing sources, although some of these primitives can be combined with our construction so to improve the output length and the error guarantee. Instead, at the heart of our construction is a new primitive called an independence-preserving merger. The construction of the latter builds on the alternating extraction technique.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/bs3ee-mwm86The duality gap for two-team zero-sum games
https://resolver.caltech.edu/CaltechAUTHORS:20190402-143205523
Authors: {'items': [{'id': 'Schulman-L-J', 'name': {'family': 'Schulman', 'given': 'Leonard J.'}, 'orcid': '0000-0001-9901-2797'}, {'id': 'Vazirani-Umesh-V', 'name': {'family': 'Vazirani', 'given': 'Umesh V.'}}]}
Year: 2017
DOI: 10.4230/LIPIcs.ITCS.2017.56
We consider multiplayer games in which the players fall in two teams of size k, with payoffs equal within, and of opposite sign across, the two teams. In the classical case of k = 1, such zero-sum games possess a unique value, independent of order of play, due to the von Neumann minimax
theorem. However, this fails for all k > 1; we can measure this failure by a duality gap, which quantifies the benefit of being the team to commit last to its strategy. In our main result we show that the gap equals 2(1 – 2^(1−k)) for m = 2 and 2(1 – m^(−(1−o(1))k)) for m > 2, with m being the
size of the action space of each player. At a finer level, the cost to a team of individual players acting independently while the opposition employs joint randomness is 1 – 2^(1−k) for k = 2, and 1 – m^(−(1−o(1))k) for m > 2.
This class of multiplayer games, apart from being a natural bridge between two-player zero-sum games and general multiplayer games, is motivated from Biology (the weak selection model of evolution) and Economics (players with shared utility but poor coordination).https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/2s0v3-paf71Convergence of Incentive-Driven Dynamics in Fisher Markets
https://resolver.caltech.edu/CaltechAUTHORS:20170214-145943460
Authors: {'items': [{'id': 'Dvijotham-Krishnamurthy', 'name': {'family': 'Dvijotham', 'given': 'Krishnamurthy'}, 'orcid': '0000-0002-1328-4677'}, {'id': 'Rabani-Yuval', 'name': {'family': 'Rabani', 'given': 'Yuval'}}, {'id': 'Schulman-L-J', 'name': {'family': 'Schulman', 'given': 'Leonard J.'}, 'orcid': '0000-0001-9901-2797'}]}
Year: 2017
DOI: 10.1137/1.9781611974782.35
In both general equilibrium theory and game theory, the dominant mathematical models rest on a fully rational solution concept in which every player's action is a best-response to the actions of the other players. In both theories there is less agreement on suitable out- of-equilibrium modeling, but one attractive approach is the level k model in which a level 0 player adopts a very simple response to current conditions, a level 1 player best-responds to a model in which others take level 0 actions, and so forth. (This is analogous to k-ply exploration of game trees in AI, and to receding-horizon control in control theory.) If players have deterministic mental models with this kind of finite-level response, there is obviously no way their mental models can all be consistent. Nevertheless, there is experimental evidence that people act this way in many situations, motivating the question of what the dynamics of such interactions lead to.
We address the problem of out-of-equilibrium price dynamics in the setting of Fisher markets. We develop a general framework in which sellers have (a) a set of atomic price update rules which are simple responses to a price vector; (b) a belief-formation procedure that simulates actions of other sellers (themselves using the atomic price updates) to some finite horizon in the future. In this framework, sellers use an atomic price update rule to respond to a price vector they generate with the belief formation procedure. The framework is general and allows sellers to have inconsistent and time- varying beliefs about each other. Under certain assumptions on the atomic update rules, we show that despite the inconsistent and time-varying nature of beliefs, the market converges to a unique equilibrium. (If the price updates are driven by weak-gross substitutes demands, this is the same equilibrium point predicted by those demands.) This result holds for both synchronous and asynchronous discrete-time updates. Moreover, the result is computationally feasible in the sense that the convergence rate is linear, i.e., the distance to equilibrium decays exponentially fast. To the best of our knowledge, this is the first result that demonstrates, in Fisher markets, convergence at any rate for dynamics driven by a plausible model of seller incentives.
We then specialize our results to Fisher markets with elastic demands (a further special case corresponds to demand generated by buyers with constant elasticity of substitution (CES) utilities, in the weak gross substitutes (WGS) regime) and show that the atomic update rule in which a seller uses the best-response (=profit- maximizing) update given the prices of all other sellers, satisfies the assumptions required on atomic price update rules in our framework. We can even characterize the convergence rate (as a function of elasticity parameters of the demand function).
Our results apply also to settings where, to the best of our knowledge, there exists no previous demonstration of efficient convergence of any discrete dynamic of price updates. Even for the simple case of (level 0) best- response dynamics, our result is the first to demonstrate a linear rate of convergence.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/b9z6j-aqc54Analysis of a Classical Matrix Preconditioning Algorithm
https://resolver.caltech.edu/CaltechAUTHORS:20170614-151934720
Authors: {'items': [{'id': 'Schulman-L-J', 'name': {'family': 'Schulman', 'given': 'Leonard J.'}, 'orcid': '0000-0001-9901-2797'}, {'id': 'Sinclair-A', 'name': {'family': 'Sinclair', 'given': 'Alistair'}}]}
Year: 2017
DOI: 10.1145/2988227
We study a classical iterative algorithm for balancing matrices in the L_∞ norm via a scaling transformation. This algorithm, which goes back to Osborne and Parlett & Reinsch in the 1960s, is implemented as a standard preconditioner in many numerical linear algebra packages. Surprisingly, despite its widespread use over several decades, no bounds were known on its rate of convergence. In this article, we prove that, for any irreducible n × n (real or complex) input matrix A, a natural variant of the algorithm converges in O(n^3 log (nρ/ϵ)) elementary balancing operations, where ρ measures the initial imbalance of A and ϵ is the target imbalance of the output matrix. (The imbalance of A is max_i | log(a_i^(out)/a_i^(in))|, where a_i^(out), a_i^(in) are the maximum entries in magnitude in the ith row and column, respectively.) This bound is tight up to the log n factor. A balancing operation scales the ith row and column so that their maximum entries are equal, and requires O(m/n) arithmetic operations on average, where m is the number of nonzero elements in A. Thus, the running time of the iterative algorithm is O(n^2m). This is the first time bound of any kind on any variant of the Osborne-Parlett-Reinsch algorithm. We also prove a conjecture of Chen that characterizes those matrices for which the limit of the balancing process is independent of the order in which balancing operations are performed.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/pjgze-8sb69Quasi-regular sequences and optimal schedules for security games
https://resolver.caltech.edu/CaltechAUTHORS:20180410-145958889
Authors: {'items': [{'id': 'Kempe-D', 'name': {'family': 'Kempe', 'given': 'David'}}, {'id': 'Schulman-L-J', 'name': {'family': 'Schulman', 'given': 'Leonard J.'}, 'orcid': '0000-0001-9901-2797'}, {'id': 'Tamuz-O', 'name': {'family': 'Tamuz', 'given': 'Omer'}, 'orcid': '0000-0002-0111-0418'}]}
Year: 2018
DOI: 10.1137/1.9781611975031.106
We study security games in which a defender commits to a mixed strategy for protecting a finite set of targets of different values. An attacker, knowing the defender's strategy, chooses which target to attack and for how long. If the attacker spends time t at a target i of value α_i, and if he leaves before the defender visits the target, his utility is t · α_i; if the defender visits before he leaves, his utility is 0. The defender's goal is to minimize the attacker's utility. The defender's strategy consists of a schedule for visiting the targets; it takes her unit time to switch between targets. Such games are a simplified model of a number of real-world scenarios such as protecting computer networks from intruders, crops from thieves, etc.
We show that optimal defender play for such security games, although played in continuous time, reduces to the solution of a combinatorial question regarding the existence of infinite sequences over a finite alphabet, with the following properties for each symbol i: (1) i constitutes a prescribed limiting fraction p_i of the sequence. (2) The occurrences of i are spread apart close to evenly, in that the ratio of the longest to shortest interval between consecutive occurrences is bounded by a parameter K. We call such sequences K-quasi-regular; a 1-quasi-regular sequence is one in which the occurrences of each symbol form an arithmetic sequence.
As we show, a 1-quasi-regular sequence ensures an optimal defender strategy for these security games: the intuition for this fact lies in the famous "inspection paradox." However, as we demonstrate, for K < 2 and general p_i, K-quasi-regular sequences may not exist. Fortunately, this does not turn out to be an obstruction: we show that, surprisingly, 2-quasi-regular sequences also suffice for optimal defender play. What is more, even randomized 2-quasi-regular sequences suffice for optimality. We show that such sequences always exist, and can be calculated efficiently. Thus, we can ensure optimal defender play for these security games.
The question of the least K for which deterministic K-quasi-regular sequences exist is fascinating. Using an ergodic theoretical approach, we proceed to show that deterministic 3-quasi-regular sequences always exist (and can be calculated efficiently). We also show that these deterministic 3-regular sequences give rise to a ≈ 1.006-approximation algorithm for the defender's optimal strategy. For 2 ≤ K < 3 we do not know whether deterministic K-quasi-regular sequences always exist; however, when the pi are all small, improved bounds are possible, and in fact, (1 + ∊)-quasi-regular deterministic sequences exist for any ∊ > 0 for sufficiently small p_i.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/x2458-79e56Explicit binary tree codes with polylogarithmic size alphabet
https://resolver.caltech.edu/CaltechAUTHORS:20180823-141633075
Authors: {'items': [{'id': 'Cohen-Gil', 'name': {'family': 'Cohen', 'given': 'Gil'}}, {'id': 'Haeupler-B', 'name': {'family': 'Haeupler', 'given': 'Bernhard'}}, {'id': 'Schulman-L-J', 'name': {'family': 'Schulman', 'given': 'Leonard J.'}, 'orcid': '0000-0001-9901-2797'}]}
Year: 2018
DOI: 10.1145/3188745.3188928
This paper makes progress on the problem of explicitly constructing a binary tree code with constant distance and constant alphabet size.
We give an explicit binary tree code with constant distance and alphabet size poly(logn), where n is the depth of the tree. This is the first improvement over a two-decade-old construction that has an exponentially larger alphabet of size poly(n).
At the core of our construction is the first explicit tree code with constant rate and constant distance, though with non-constant arity - a result of independent interest. This construction adapts the polynomial interpolation framework to the online setting.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/c1fyz-btw70Quasi-random multilinear polynomials
https://resolver.caltech.edu/CaltechAUTHORS:20190103-132542769
Authors: {'items': [{'id': 'Kalai-Gil', 'name': {'family': 'Kalai', 'given': 'Gil'}, 'orcid': '0000-0003-0982-1000'}, {'id': 'Schulman-L-J', 'name': {'family': 'Schulman', 'given': 'Leonard J.'}, 'orcid': '0000-0001-9901-2797'}]}
Year: 2019
DOI: 10.1007/s11856-018-1821-y
We consider multilinear Littlewood polynomials, polynomials in n variables in which a specified set of monomials U have ±1 coefficients, and all other coefficients are 0. We provide upper and lower bounds (which are close for U of degree below log n) on the minimum, over polynomials h consistent with U, of the maximum of |h| over ±1 assignments to the variables. (This is a variant of a question posed by Erdős regarding the maximum on the unit disk of univariate polynomials of given degree with unit coefficients.) We outline connections to the theory of quasi-random graphs and hypergraphs, and to statistical mechanics models. Our methods rely on the analysis of the Gale–Berlekamp game; on the constructive side of the generic chaining method; on a Khintchine-type inequality for polynomials of degree greater than 1; and on Bernstein's approximation theory inequality.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/6wn6k-w7j96The duality gap for two-team zero-sum games
https://resolver.caltech.edu/CaltechAUTHORS:20190328-112416478
Authors: {'items': [{'id': 'Schulman-L-J', 'name': {'family': 'Schulman', 'given': 'Leonard J.'}, 'orcid': '0000-0001-9901-2797'}, {'id': 'Vazirani-Umesh-V', 'name': {'family': 'Vazirani', 'given': 'Umesh V.'}}]}
Year: 2019
DOI: 10.1016/j.geb.2019.03.011
We consider multiplayer games in which the players fall in two teams of size k, with payoffs equal within, and of opposite sign across, the two teams. In the classical case of k = 1, such zero-sum games possess a unique value, independent of order of play. However, this fails for all k > 1; we can measure this failure by a duality gap, which quantifies the benefit of being the team to commit last to its strategy. We show that the gap equals 2(1−2^(1−k)) for m = 2 and 2(1−m^(−(1−o(1))k)) for m > 2, with m being the size of the action space of each player. Extensions hold also for different-size teams and players with various-size action spaces.
We further study the effect of exchanging order of commitment among individual players (not only among the entire teams).
The class of two-team zero-sum games is motivated from the weak selection model of evolution, and from considering teams such as firms in which independent players (ideally) have shared utility.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/megb3-qq676Online Codes for Analog Signals
https://resolver.caltech.edu/CaltechAUTHORS:20191010-093119072
Authors: {'items': [{'id': 'Schulman-L-J', 'name': {'family': 'Schulman', 'given': 'Leonard J.'}, 'orcid': '0000-0001-9901-2797'}, {'id': 'Srivastava-P', 'name': {'family': 'Srivastava', 'given': 'Piyush'}, 'orcid': '0000-0003-0953-2890'}]}
Year: 2019
DOI: 10.1109/TIT.2019.2919632
This paper revisits a classical scenario in communication theory: a waveform sampled at regular intervals is to be encoded so as to minimize distortion in its reconstruction, despite the noise. This transformation must be online (causal), to enable real-time signaling, and should use no more power than the original signal. The noise model we consider is an atomic norm convex relaxation of the standard (discrete alphabet) Hammingweight-bounded model, namely adversarial ℓ_1 -bounded. In the block coding (noncausal) setting, such encoding is possible due to the existence of large almost-Euclidean sections in ℓ_1 spaces, a notion first studied in the work of Dvoretzky in 1961. Our main result is that an analogous result is achievable even casually. Equivalently, our work may be seen as a lower triangular version of ℓ_1 Dvoretzky theorems. In terms of communication, the guarantees are expressed in terms of certain time-weighted norms: the time-weighted ℓ_2 norm imposed on the decoder forces increasingly accurate reconstruction of the distant past signal, while the time-weighted ℓ_1 norm on the noise ensures vanishing interference from distant past noise. Encoding is linear (hence easy to implement in analog hardware). Decoding is performed by an LP analogous to those used in compressed sensing.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/dfz5s-6q897Achieving target equilibria in network routing games without knowing the latency functions
https://resolver.caltech.edu/CaltechAUTHORS:20180622-082816994
Authors: {'items': [{'id': 'Bhaskar-U', 'name': {'family': 'Bhaskar', 'given': 'Umang'}}, {'id': 'Ligett-K', 'name': {'family': 'Ligett', 'given': 'Katrina'}, 'orcid': '0000-0003-2780-6656'}, {'id': 'Schulman-L-J', 'name': {'family': 'Schulman', 'given': 'Leonard J.'}, 'orcid': '0000-0001-9901-2797'}, {'id': 'Swamy-C', 'name': {'family': 'Swamy', 'given': 'Chaitanya'}}]}
Year: 2019
DOI: 10.1016/j.geb.2018.02.009
The analysis of network routing games typically assumes precise, detailed information about the latency functions. Such information may, however, be unavailable or difficult to obtain. Moreover, one is often primarily interested in enforcing a desired target flow as an equilibrium. We ask whether one can achieve target flows as equilibria without knowing the underlying latency functions. We give a crisp positive answer to this question. We show that one can efficiently compute edge tolls that induce a given target multicommodity flow in a nonatomic routing game using a polynomial number of queries to an oracle that takes tolls as input and outputs the resulting equilibrium flow. This result is obtained via a novel application of the ellipsoid method, and extends to various other settings. We obtain improved query-complexity bounds for series-parallel networks, and single-commodity routing games with linear latency functions. Our techniques provide new insights into network routing games.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/zyfqc-8bf55Edge Expansion and Spectral Gap of Nonnegative Matrices
https://resolver.caltech.edu/CaltechAUTHORS:20200828-114509192
Authors: {'items': [{'id': 'Mehta-J-C', 'name': {'family': 'Mehta', 'given': 'Jenish C.'}}, {'id': 'Schulman-L-J', 'name': {'family': 'Schulman', 'given': 'Leonard J.'}, 'orcid': '0000-0001-9901-2797'}]}
Year: 2020
DOI: 10.1137/1.9781611975994.73
The classic graphical Cheeger inequalities state that if M is an n × n symmetric doubly stochastic matrix, then
1−λ₂(M)/2 ≤ ϕ(M) ≤ √2⋅(1−λ₂(M)) where ϕ(M) = min_(S⊆[n],|S|≤n/2)(1|S|∑_(i∈S,j∉S)M_(i,j)) is the edge expansion of M, and λ₂(M) is the second largest eigenvalue of M. We study the relationship between φ(A) and the spectral gap 1 – Re λ₂(A) for any doubly stochastic matrix A (not necessarily symmetric), where λ₂(A) is a nontrivial eigenvalue of A with maximum real part. Fiedler showed that the upper bound on φ(A) is unaffected, i.e., ϕ(A) ≤ √2⋅(1−Reλ₂(A)). With regards to the lower bound on φ(A), there are known constructions with ϕ(A) ∈ Θ(1−Reλ₂(A)/log n) indicating that at least a mild dependence on n is necessary to lower bound φ(A). In our first result, we provide an exponentially better construction of n × n doubly stochastic matrices A_n, for which ϕ(An) ≤ 1−Reλ₂(A_n)/√n. In fact, all nontrivial eigenvalues of our matrices are 0, even though the matrices are highly nonexpanding. We further show that this bound is in the correct range (up to the exponent of n), by showing that for any doubly stochastic matrix A, ϕ(A) ≥ 1−Reλ₂(A)/35⋅n As a consequence, unlike the symmetric case, there is a (necessary) loss of a factor of n^α for ½ ≤ α ≤ 1 in lower bounding φ by the spectral gap in the nonsymmetric setting. Our second result extends these bounds to general matrices R with nonnegative entries, to obtain a two-sided gapped refinement of the Perron-Frobenius theorem. Recall from the Perron-Frobenius theorem that for such R, there is a nonnegative eigenvalue r such that all eigenvalues of R lie within the closed disk of radius r about 0. Further, if R is irreducible, which means φ(R) > 0 (for suitably defined φ), then r is positive and all other eigenvalues lie within the open disk, so (with eigenvalues sorted by real part), Re λ₂(R) < r. An extension of Fiedler's result provides an upper bound and our result provides the corresponding lower bound on φ(R) in terms of r – Re λ₂(R), obtaining a two-sided quantitative version of the Perron-Frobenius theorem.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/nmy3x-1w272The invisible hand of Laplace: The role of market structure in price convergence and oscillation
https://resolver.caltech.edu/CaltechAUTHORS:20210128-142707495
Authors: {'items': [{'id': 'Rabani-Yuval', 'name': {'family': 'Rabani', 'given': 'Yuval'}}, {'id': 'Schulman-L-J', 'name': {'family': 'Schulman', 'given': 'Leonard J.'}, 'orcid': '0000-0001-9901-2797'}]}
Year: 2021
DOI: 10.1016/j.jmateco.2021.102475
The "invisible hand" of the free market is remarkably effective at producing near-equilibrium prices. This is difficult to quantify, however, in the absence of an agreed model for out-of-equilibrium trade. Short of a fully reductionist model, a useful substitute would be a scaling law relating equilibration time and other market parameters. Even this, however, is missing in the literature.
We make progress in this direction. We examine a class of Arrow–Debreu markets with price signaling driven by continuous-time proportional-tâtonnement. We show that the connectivity among the participants in the market determines quite accurately a scaling law for convergence time of the market to equilibrium, and thus determines the effectiveness of the price signaling. To our knowledge this is the first characterization of price stability in terms of market connectivity. At a technical level, we show how convergence in our class of markets is determined by a market-dependent Laplacian matrix.
If a market is not isolated but, rather, subject to external noise, equilibrium theory has predictive value only to the extent to which that noise is counterbalanced by the price equilibration process. Our model quantifies this predictive value by providing a scaling law that relates the connectivity of the market with the variance of its prices.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/c52mf-8e731Hadamard Extensions and the Identification of Mixtures of Product Distributions
https://resolver.caltech.edu/CaltechAUTHORS:20220210-721777000
Authors: {'items': [{'id': 'Gordon-Spencer-L', 'name': {'family': 'Gordon', 'given': 'Spencer L.'}}, {'id': 'Schulman-L-J', 'name': {'family': 'Schulman', 'given': 'Leonard J.'}, 'orcid': '0000-0001-9901-2797'}]}
Year: 2022
DOI: 10.1109/tit.2022.3146630
The Hadamard Extension H(m) of an n×k matrix m is the collection of all Hadamard products of subsets of its rows. This construction is essential for source identification (parameter estimation) of a mixture of k product distributions over n binary random variables. A necessary requirement for such identification is that H(m) have full column rank; conversely, identification is possible if apart from each row there exist two disjoint sets of rows of m, each of whose extension has full column rank. It is necessary therefore to understand when H(m) has full column rank; we provide two results in this direction. The first is that if H(m) has full column rank then there exists a set of at most k−1 rows of m, whose extension already has full column rank. The second is a Hall-type condition on the values in the rows of m, that suffices to ensure full column rank of H(m).https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/sjy55-zw036A refined approximation for Euclidean k-means
https://resolver.caltech.edu/CaltechAUTHORS:20220204-680165000
Authors: {'items': [{'id': 'Grandoni-Fabrizio', 'name': {'family': 'Grandoni', 'given': 'Fabrizio'}, 'orcid': '0000-0002-9676-4931'}, {'id': 'Ostrovsky-Rafail', 'name': {'family': 'Ostrovsky', 'given': 'Rafail'}}, {'id': 'Rabani-Yuval', 'name': {'family': 'Rabani', 'given': 'Yuval'}, 'orcid': '0000-0001-7772-2544'}, {'id': 'Schulman-L-J', 'name': {'family': 'Schulman', 'given': 'Leonard J.'}, 'orcid': '0000-0001-9901-2797'}, {'id': 'Venkar-Rakesh', 'name': {'family': 'Venkat', 'given': 'Rakesh'}}]}
Year: 2022
DOI: 10.1016/j.ipl.2022.106251
In the Euclidean k-Means problem we are given a collection of n points D in an Euclidean space and a positive integer k. Our goal is to identify a collection of k points in the same space (centers) so as to minimize the sum of the squared Euclidean distances between each point in D and the closest center. This problem is known to be APX-hard and the current best approximation ratio is a primal-dual 6.357 approximation based on a standard LP for the problem [Ahmadian et al. FOCS'17, SICOMP'20].
In this note we show how a minor modification of Ahmadian et al.'s analysis leads to a slightly improved 6.12903 approximation. As a related result, we also show that the mentioned LP has integrality gap at least (16+5√)/15 > 1.2157.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/yb16t-5cd76Convergence of incentive-driven dynamics in Fisher markets
https://resolver.caltech.edu/CaltechAUTHORS:20201217-133744650
Authors: {'items': [{'id': 'Dvijotham-Krishnamurthy', 'name': {'family': 'Dvijotham', 'given': 'Krishnamurthy'}, 'orcid': '0000-0002-1328-4677'}, {'id': 'Rabani-Yuval', 'name': {'family': 'Rabani', 'given': 'Yuval'}, 'orcid': '0000-0001-7772-2544'}, {'id': 'Schulman-L-J', 'name': {'family': 'Schulman', 'given': 'Leonard J.'}, 'orcid': '0000-0001-9901-2797'}]}
Year: 2022
DOI: 10.1016/j.geb.2020.11.005
We study out-of-equilibrium price dynamics in Fisher markets. We develop a general framework in which sellers have (a) a set of atomic price update rules (APU), which are simple responses to a price vector; (b) a belief-formation procedure that simulates actions of other sellers (themselves using the APU) to some finite horizon in the future. Sellers use an APU to respond to a price vector they generate with the belief formation procedure. The framework allows sellers to have inconsistent and time-varying beliefs about each other. Under mild and natural assumptions on the APU, we show that despite the inconsistent and time-varying nature of beliefs, the market converges to a unique equilibrium at a linear rate (distance to equilibrium decreases exponentially in time). If the APU are driven by weak gross substitutes demands, the equilibrium point is the same as predicted by those demands.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/3wmyd-8sk65