Article records
https://feeds.library.caltech.edu/people/Schulman-L-J/article.rss
A Caltech Library Repository Feedhttp://www.rssboard.org/rss-specificationpython-feedgenenFri, 12 Apr 2024 14:16:18 +0000Coding 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.edu/records/1gk21-mpy27Signal 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.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.edu/records/0h454-kks87A 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.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.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.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.edu/records/5sxrn-qcg43Quantum 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.edu/records/np257-djj69A 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.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.edu/records/bcrvh-peb49Physical 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.edu/records/7rbaq-q5m53A 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.edu/records/9fta2-dhr34Convergence 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.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.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.edu/records/qh2xd-5m710Lower 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.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.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.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.edu/records/8gh3w-2b995The 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.edu/records/c3f6z-nhz24Analysis 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.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.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.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.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.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.edu/records/ypn80-6m171Clustering 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.edu/records/yhwnt-q8v98The 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.edu/records/hw9y3-6q494The 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.edu/records/tewn0-mep45Optimal 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.edu/records/tmydw-2gs51Optimal 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.edu/records/9zea3-qk087Dimension-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.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.edu/records/qahjw-gta16Analysis 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.edu/records/pjgze-8sb69Quasi-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.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.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.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.edu/records/zyfqc-8bf55The 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.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.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.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.edu/records/3wmyd-8sk65