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A Caltech Library Repository Feedhttp://www.rssboard.org/rss-specificationpython-feedgenenTue, 16 Apr 2024 15:57:52 +0000Automating Resource Management for Distributed Business Processes
https://resolver.caltech.edu/CaltechETD:etd-11012005-093745
Authors: {'items': [{'email': 'ginis@alumni.caltech.edu', 'id': 'Ginis-Roman', 'name': {'family': 'Ginis', 'given': 'Roman'}, 'show_email': 'YES'}]}
Year: 2002
DOI: 10.7907/9GXT-BD03
A distributed business process is a set of related activities performed by independent resources offering services for lease. For instance, constructing an office building involves hundreds of activities such as excavating, plumbing and carpentry performed by machines and subcontractors, whose activities are related in time, space, cost and other dimensions. In the last decade Internet-based middleware has linked consumers with resources and services enabling the consumers to more efficiently locate, select and reserve the resources for use in business processes. This recent capability creates an opportunity for a new automation of resource management that can assign the optimal resources to the activities of a business process to maximize its utility to the consumer and yield substantial gains in operational efficiency.
This thesis explores two basic problems towards automating the management of distributed business processes: 1. How to choose the best resources for the activities of a process (the Activity Resource Assignment - ARA - optimization problem); and 2. How to reserve the resources chosen for a process as an atomic operation when time has value, i.e., commit all resources or no resources (the Distributed Service Commit problem - DSC). I believe these will become the typical optimization and agreement problems between consumers and producers in a networked service economy.
I propose a solution to the ARA optimization problem by modeling it as a special type of Integer Programming and I give a method for solving it efficiently for a large class of practical cases. Given a problem instance the method extracts the structure of the problem and using a new concept of variable independence recursively simplifies it while retaining at least one optimal solution. The reduction operation is guided by a novel procedure that makes use of the recent advances in tree-decomposition of graphs from the graph complexity theory.
The solution to the DSC problem is an algorithm based on financial instruments and the two-phase commit protocol adapted for services. The method achieves an economically sensible atomic reservation agreement between multiple distributed resources and consumers in a free market environment.
I expect the automation of resource management addressed in my thesis and elsewhere will pave the way for more efficient business operations in the networked economy.https://thesis.library.caltech.edu/id/eprint/4357Wireless Networks, from Collective Behavior to the Physics of Propagation
https://resolver.caltech.edu/CaltechETD:etd-05202003-154451
Authors: {'items': [{'email': 'massimo@ece.ucsd.edu', 'id': 'Franceschetti-Massimo', 'name': {'family': 'Franceschetti', 'given': 'Massimo'}, 'orcid': '0000-0002-4057-8152', 'show_email': 'YES'}]}
Year: 2003
DOI: 10.7907/SCTG-FN57
This thesis addresses some of the key challenges in the emerging wireless scenario. It focuses on the problems of connectivity, coverage, and wave propagation, following a mathematically rigorous approach. The questions addressed are very basic and extremely easy to state. Their solution, however, can be difficult and leads to the development of a new kind of percolation theory, to a new theorem in geometry, and to a new model of wave propagation in urban environments. The problems are connected together to provide guidelines in the design of wireless networks.
https://thesis.library.caltech.edu/id/eprint/1887Networks of Relations
https://resolver.caltech.edu/CaltechETD:etd-06032005-140944
Authors: {'items': [{'email': 'cook@ini.uzh.ch', 'id': 'Cook-Matthew-M', 'name': {'family': 'Cook', 'given': 'Matthew M.'}, 'show_email': 'NO'}]}
Year: 2005
DOI: 10.7907/CVKM-D684
<p>Relations are everywhere. In particular, we think and reason in terms of mathematical and English sentences that state relations. However, we teach our students much more about how to manipulate functions than about how to manipulate relations. Consider functions. We know how to combine functions to make new functions, how to evaluate functions efficiently, and how to think about compositions of functions. Especially in the area of boolean functions, we have become experts in the theory and art of designing combinations of functions to yield what we want, and this expertise has led to techniques that enable us to implement mind-bogglingly large yet efficient networks of such functions in hardware to help us with calculations. If we are to make progress in getting machines to be able to reason as well as they can calculate, we need to similarly develop our understanding of relations, especially their composition, so we can develop techniques to help us bridge between the large and small scales. There has been some important work in this area, ranging from practical applications such as relational databases to extremely theoretical work in universal algebra, and sometimes theory and practice manage to meet, such as in the programming language Prolog, or in the probabilistic reasoning methods of artificial intelligence. However, the real adventure is yet to come, as we learn to develop a better understanding of how relations can efficiently and reliably be composed to get from a low level representation to a high level representation, as this understanding will then allow the development of automated techniques to do this on a grand scale, finally enabling us to build machines that can reason as amazingly as our contemporary machines can calculate.</p>
<p>This thesis explores new ground regarding the composition of relations into larger relational structures. First of all a foundation is laid by examining how networks of relations might be used for automated reasoning. We define exclusion networks, which have close connections with the areas of constraint satisfaction problems, belief propagation, and even boolean circuits. The foundation is laid somewhat deeper than usual, taking us inside the relations and inside the variables to see what is the simplest underlying structure that can satisfactorily represent the relationships contained in a relational network. This leads us to define zipper networks, an extremely low-level view in which the names of variables or even their values are no longer necessary, and relations and variables share a common substrate that does not distinguish between the two. A set of simple equivalence operations is found that allows one to transform a zipper network while retaining its solution structure, enabling a relation-variable duality as well as a canonical form on linear segments. Similarly simple operations allow automated deduction to take place, and these operations are simple and uniform enough that they are easy to imagine being implemented by biological neural structures.</p>
<p>The canonical form for linear segments can be represented as a matrix, leading us to matrix networks. We study the question of how we can perform a change of basis in matrix networks, which brings us to a new understanding of Valiant's recent holographic algorithms, a new source of polynomial time algorithms for counting problems on graphs that would otherwise appear to take exponential time. We show how the holographic transformation can be understood as a collection of changes of basis on individual edges of the graph, thus providing a new level of freedom to the method, as each edge may now independently choose a basis so as to transform the matrices into the required form.</p>
<p>Consideration of zipper networks makes it clear that "fan-out," i.e., the ability to duplicate information (for example allowing a variable to be used in many places), is most naturally itself represented as a relation along with everything else. This is a notable departure from the traditional lack of representation for this ability. This deconstruction of fan-out provides a more general model for combining relations than was provided by previous models, since we can examine both the traditional case where fan-out (the equality relation on three variables) is available and the more interesting case where its availability is sub ject to the same limitations as the availability of other relations. As we investigate the composition of relations in this model where fan-out is explicit, what we find is very different from what has been found in the past.</p>
<p>First of all we examine the relative expressive power of small relations: For each relation on three boolean variables, we examine which others can be implemented by networks built solely from that relation. (We also find, in each of these cases, the complexity of deciding whether such a network has a solution. We find that solutions can be found in polynomial time for all but one case, which is NP-complete.) For the question of which relations are able to implement which others, we provide an extensive and complete answer in the form of a hierarchy of relative expressive power for these relations. The hierarchy for relations is more complex than Post's well-known comparable hierarchy for functions, and parts of it are particularly difficult to prove. We find an explanation for this phenomenon by showing that in fact, the question of whether one relation can implement another (and thus should be located above it in the hierarchy) is undecidable. We show this by means of a complicated reduction from the halting problem for register machines. The hierarchy itself has a lot of structure, as it is rarely the case that two ternary boolean relations are equivalent. Often they are comparable, and often they are incomparable—the hierarchy has quite a bit of width as well as depth. Notably, the fan-out relation is particularly difficult to implement; only a very few relations are capable of implementing it. This provides an additional ex post facto justification for considering the case where fan-out is absent: If you are not explicitly provided with fan-out, you are unlikely to be able to implement it.</p>
<p>The undecidability of the hierarchy contrasts strongly with the traditional case, where the ubiquitous availability of fan-out causes all implementability questions to collapse into a finite decidable form. Thus we see that for implementability among relations, fan-out leads to undecidability. We then go on to examine whether this result might be taken back to the world of functions to find a similar difference there. As we study the implementability question among functions without fan-out, we are led directly to questions that are independently compelling, as our functional implementability question turns out to be equivalent to asking what can be computed by sets of chemical reactions acting on a finite number of species. In addition to these chemical reaction networks, several other nondeterministic systems are also found to be equivalent in this way to the implementability question, namely, Petri nets, unordered Fractran, vector addition systems, and "broken" register machines (whose decrement instruction may fail even on positive registers). We prove equivalences between these systems.</p>
<p>We find several interesting results in particular for chemical reaction networks, where the standard model has reaction rates that depend on concentration. In this setting, we analyze questions of possibility as well as questions of probability. The question of the possibility of reaching a target state turns out to be equivalent to the reachability question for Petri nets and vector addition systems, which has been well studied. We provide a new proof that a form of this reachability question can be decided by primitive recursive functions. Ours is the first direct proof of this relationship, avoiding the traditional excursion to Diophantine equations, and thus providing a crisper picture of the relationship between Karp's coverability tree and primitive recursive functions.</p>
<p>In contrast, the question of finding the probability (according to standard chemical kinetics) of reaching a given target state turns out to be undecidable. Another way of saying this is that if we wish to distinguish states with zero probability of occurring from states with positive probability of occurring, we can do so, but if we wish to distinguish low probability states from high probability states, there is no general way to do so. Thus, if we wish to use a chemical reaction network to perform a computation, then if we insist that the network must always get the right answer, we will only be able to use networks with limited computational power, but if we allow just the slightest probability of error, then we can use networks with Turing-universal computational ability. This power of probability is quite surprising, especially when contrasted with the conventional computational complexity belief that BPP = P.</p>
<p>Exploring the source of this probabilistic power, we find that the probabilities guiding the network need to depend on the concentrations (or perhaps on time)—fixed probabilities aren’t enough on their own to achieve this power. In the language of Petri nets, if one first picks a transition at random, and then fires it if it is enabled, then the probability of reaching a particular target state can be calculated to arbitrary precision, but if one first picks a token at random, and then fires an enabled transition that will absorb that token, then the probability of reaching a particular target state cannot in general be calculated to any precision whatsoever.</p>
<p>In short, what started as a simple thorough exploration of the power of composition of relations has led to many decidability and complexity questions that at first appear completely unrelated, but turn out to combine to paint a coherent picture of the relationship between relations and functions, implementability and reachability, possibility and probability, and decidability and undecidability.</p>https://thesis.library.caltech.edu/id/eprint/2424Design and Analysis of Network Codes
https://resolver.caltech.edu/CaltechETD:etd-05302006-131149
Authors: {'items': [{'email': 'jaggi@ie.cuhk.edu.hk', 'id': 'Jaggi-Sidharth', 'name': {'family': 'Jaggi', 'given': 'Sidharth'}, 'orcid': '0000-0002-5522-7486', 'show_email': 'YES'}]}
Year: 2006
DOI: 10.7907/7ERZ-H253
<p>The information theoretic aspects of large networks with many terminals present several interesting and non-intuitive phenomena. One such crucial phenomenon was first explored in a detailed manner in the excellent work by Ahlswede at al. It compared two paradigms for operating a network -- one in which interior nodes were restricted to only copying and forwarding incoming messages on outgoing links, and another in which internal nodes were allowed to perform non-trivial arithmetic operations on information on incoming links to generate information on outgoing links. It showed that the latter approach could substantially improve throughput compared to the more traditional scenario. Further work by various authors showed how to design codes (called network codes) to transmit under this new paradigm and also demonstrated exciting new phenomena for these codes such as robustness against network failures, distributed design, and increased security.</p>
<p>In this work, we consider the low-complexity design and analysis of network codes, with a focus on codes for multicasting information. We examine both centralized and decentralized design of such codes, and also both randomized and deterministic design algorithms. We compare different notions of linearity and show the interplay between these notions in the design of linear network codes. We determine bounds on the complexity of network codes. We also consider the problem of error-correction and secrecy for network codes when a malicious adversary controls some subset of the network resources.</p>https://thesis.library.caltech.edu/id/eprint/2304Distributed Estimation and Control in Networked Systems
https://resolver.caltech.edu/CaltechETD:etd-08172006-130145
Authors: {'items': [{'id': 'Gupta-Vijay', 'name': {'family': 'Gupta', 'given': 'Vijay'}, 'show_email': 'NO'}]}
Year: 2007
DOI: 10.7907/KWN2-X741
<p>Rapid advances in information processing, communication and sensing technologies have enabled more and more devices to be provided with embedded processors, networking capabilities and sensors. For the field of estimation and control, it is now possible to consider an architecture in which many simple components communicate and cooperate to achieve a joint team goal. This distributed (or networked) architecture promises much in terms of performance, reliability and simplicity of design; however, at the same time, it requires extending the traditional theories of control, communication and computation and, in fact, looking at a unified picture of the three fields. A systematic theory of how to design distributed systems is currently lacking.</p>
<p>This dissertation takes the first steps towards understanding the effects of imperfect information flow in distributed systems from an estimation and control perspective and coming up with new design principles to counter these effects. Designing networked systems is difficult because such systems challenge two basic assumptions of traditional control theory - presence of a central node with access to all the information about the system and perfect transmission of information among components. We formulate and solve many problems that deal with the removal of one, or both, of these assumptions. The chief idea explored in this dissertation is the joint design of information flow and the control law. While traditional control design has concentrated on calculating the optimal control input by assuming a particular information flow between the components, our approach seeks to synthesize the optimal information flow along with the optimal control law that satisfies the constraints of the information flow. Thus besides the question of 'What should an agent do?', the questions of 'Whom should an agent talk to?', 'What should an agent communicate?', 'When should an agent communicate?' and so on also have to be answered. The design of the information flow represents an important degree of freedom available to the system designer that has hitherto largely been ignored. As we demonstrate in the dissertation, the joint design of information flow and the optimal control input satisfying the constraints of that information flow yields large improvements in performance over simply trying to fit traditional design theories on distributed systems.</p>
<p>We begin by formulating a distributed control problem in which many agents in a formation need to cooperate to minimize a joint cost function. We provide numerical algorithms to synthesize the optimal constrained control law that involve solving linear equations only and hence are free from numerical issues plaguing the other approaches proposed in the literature. We then provide and analyze a model to understand the issue of designing the topology according to which the agents interact. The results are very surprising since there are cases when allowing communication to happen between two agents may, in fact, be detrimental to the performance.</p>
<p>We then move on to consider the effects of communication channels on control performance. To counter such effects, we propose the idea of encoding information for the purpose of estimation and control prior to transmission. Although information theoretic techniques are not possible in our problem, we are able to solve for a recursive yet optimal encoder / decoder structure in many cases. This information flow design oriented approach has unique advantages such as being optimal for any packet drop pattern, being able to include the effect of known but random delays easily, letting us escape the limits set by reliability for transmission of data across a network by using intermediate nodes as 'repeaters' similar to a digital communication network and so on.</p>
<p>We finally take a look at combining the effects of multiple sources of information and communication channels on estimation and control. We look at a distributed estimation problem in which, at every time step, only a subset out of many sensors can transmit information to the estimator. This is also a representative resource allocation problem. We propose the idea of stochastic communication patterns that allows us to include the effects of communication channels explicitly during system design. Thus, instead of tree-search based algorithms proposed in the literature, we provide stochastic scheduling algorithms that can take into account the random packet drop effect of the channels. We also consider a distributed control problem with switching topologies and solve for the optimal controller. The tools that we develop are applicable to many other scenarios and we demonstrate some of them in the dissertation.</p>
<p>Along the way, we look at many other related problems in the dissertation. As an example, we provide initial results about the issue of robustness of a distributed system design to a malfunctioning agent. This notion is currently lacking in the control and estimation community, but has to be a part of any effective theory for designing networked or distributed systems.</p>https://thesis.library.caltech.edu/id/eprint/3157Wireless Networks: New Models and Results
https://resolver.caltech.edu/CaltechETD:etd-10032006-113124
Authors: {'items': [{'id': 'Gowaikar-Radhika', 'name': {'family': 'Gowaikar', 'given': 'Radhika'}, 'show_email': 'NO'}]}
Year: 2007
DOI: 10.7907/9652-QK53
<p>Wireless communications have gained much currency in the last few decades. In this thesis we present results regarding several wireless communication systems, in particular, wireless networks.</p>
<p>For some time now, it has been known that in an ad hoc network in which nodes share the wireless medium, and the connection strengths between nodes follow a distance-based decay law, the throughput scales like O(√n), where n is the number of nodes. In Chapter 2 we introduce randomness in the connection strengths and examine the effects of this on the throughput. We assume that all the channels are drawn independently from a common distribution and are not governed by a distance-decay law. It turns out that the aggregate information flow depends strongly on the distribution from which the channel strengths are drawn. For certain distributions, a throughput of n/(log n)<sup>d</sup> with d>0 is possible, which is a significant improvement over the O(√n) results known previously. In Chapter 3, we generalize the network model to two-scale networks. This model incorporates the distance-decay law for nodes that are separated by large distances, while maintaining randomness in close neighborhoods of a node. For certain networks, we show that a throughput of the form n<sup>1/t-1</sup>/log<sup>2</sup>n is achievable, where t>2 is a parameter that depends on the distribution of the connection at the local scale and is independent of the decay law that operates at a global scale.</p>
<p>In Chapter 4, we consider a model of an erasure wireless network, in which every node is connected to certain other nodes by erasure links, on which packets or bits are lost with some probability and received accurately otherwise. Each node is constrained to send the same message on all outgoing channels, thus incorporating the broadcast feature, and we assume that there is no interference in the network, other than through the possible correlation of erasure occurrences. For such networks and in certain multicast scenarios, we obtain the precise capacity region. This region has a nice max-flow, min-cut interpretation and can be achieved using linear codes. We do require the side-information regarding erasure locations on all links to be available to the destinations. Thus, we have the capacity region for a non-trivial class of wireless networks.</p>
<p>Recent results for wireline networks show that in several scenarios, it is optimal to operate these networks by making each link error-free. In Chapter 5, we consider Gaussian networks with broadcast and interference, and erasure networks with broadcast, and show that in the presence of these wireless features, it is suboptimal to make each link or sub-network error-free. We then consider these networks with the constraint that each node is permitted to either retransmit the received information or decode it and retransmit the original source information. We propose a greedy algorithm that determines the optimal operation for each node, such that the rate achievable at the destination is maximized. Further, we present decentralized implementations of this algorithm that allow each node to determine for itself the optimal operation that it needs to perform.</p>
<p>In Chapter 6, we consider a point-to-point communication system, involving multiple antennas at the transmitter and the receiver. These systems can give high data rates provided we can perform optimum, or maximum-likelihood, decoding of the received message. This problem typically reduces to that of finding the lattice point closest to a given point x in N-dimensional space. This is an integer least-squares problem and is NP-complete. The sphere decoder is an algorithm that performs decoding in an efficient manner by searching for the closest point only within a spherical region around x. In Chapter 6, we propose an algorithm that performs decoding in a sub-optimal manner by pruning the search region based on the statistics of the problem. This algorithm offers significant computational savings relative to the sphere decoder and allows us to tradeoff performance with computational complexity. Bounds on the error performance as well the complexity are presented.</p>https://thesis.library.caltech.edu/id/eprint/3885Limited Randomness in Games, and Computational Perspectives in Revealed Preference
https://resolver.caltech.edu/CaltechETD:etd-06042009-233839
Authors: {'items': [{'email': 'shankar@cs.caltech.edu', 'id': 'Kalyanaraman-Shankar', 'name': {'family': 'Kalyanaraman', 'given': 'Shankar'}, 'show_email': 'NO'}]}
Year: 2009
DOI: 10.7907/KH85-HJ73
<p>In this dissertation, we explore two particular themes in connection with the study of games and general economic interactions: bounded resources and rationality. The rapidly maturing field of algorithmic game theory concerns itself with looking at the computational limits and effects when agents in such an interaction make choices in their "self-interest." The solution concepts that have been studied in this regard, and which we shall focus on in this dissertation, assume that agents are capable of randomizing over their set of choices. We posit that agents are randomness-limited in addition to being computationally bounded, and determine how this affects their equilibrium strategies in different scenarios.</p>
<p>In particular, we study three interpretations of what it means for agents to be randomness-limited, and offer results on finding (approximately) optimal strategies that are randomness-efficient:<br />
1. One-shot games with access to the support of the optimal strategies: for this case, our results are obtained by sampling strategies from the optimal support by performing a random walk on an expander graph.<br />
2. Multiple-round games where agents have no a priori knowledge of their payoffs: we significantly improve the randomness-efficiency of known online algorithms for such games by utilizing distributions based on almost pairwise independent random variables.<br />
3. Low-rank games: for games in which agents' payoff matrices have low rank, we devise "fixed-parameter" algorithms that compute strategies yielding approximately optimal payoffs for agents, and are polynomial-time in the size of the input and the rank of the payoff tensors.</p>
<p>In regard to rationality, we look at some computational questions in a related line of work known as revealed preference theory, with the purpose of understanding the computational limits of inferring agents' payoffs and motives when they reveal their preferences by way of how they act. We investigate two problem settings as applications of this theory and obtain results about their intractability:<br />
1. Rationalizability of matchings: we consider the problem of rationalizing a given collection of bipartite matchings and show that it is NP-hard to determine agent preferences for which matchings would be stable. Further, we show, assuming P ≠ NP, that this problem does not admit polynomial-time approximation schemes under two suitably defined notions of optimization.<br />
2. Rationalizability of network formation games: in the case of network formation games, we take up a particular model of connections known as the Jackson-Wolinsky model in which nodes in a graph have valuations for each other and take their valuations into consideration when they choose to build edges. We show that under a notion of stability, known as pairwise stability, the problem of finding valuations that rationalize a collection of networks as pairwise stable is NP-hard. More significantly, we show that this problem is hard even to approximate to within a factor 1/2 and that this is tight.</p>
<p>Our results on hardness and inapproximability of these problems use well-known techniques from complexity theory, and particularly in the case of the inapproximability of rationalizing network formation games, PCPs for the problem of satisfying the optimal number of linear equations in positive integers, building on recent results of Guruswami and Raghavendra.</p>
https://thesis.library.caltech.edu/id/eprint/5267On Matrix Factorization and Scheduling for Finite-Time Average-Consensus
https://resolver.caltech.edu/CaltechTHESIS:05022010-193157687
Authors: {'items': [{'email': 'kokevin@gmail.com', 'id': 'Ko-Chih-Kai', 'name': {'family': 'Ko', 'given': 'Chih-Kai'}, 'show_email': 'NO'}]}
Year: 2010
DOI: 10.7907/GCT7-5Y66
We study the problem of communication scheduling for finite-time average-consensus in arbitrary connected networks. Viewing this consensus problem as a factorization of 1/n 11<sup>T</sup> by network-admissible families of matrices, we prove the existence of finite factorizations, provide scheduling algorithms for finite-time average consensus, and derive almost tight lower bounds on the size of the minimal factorization.https://thesis.library.caltech.edu/id/eprint/5763Limits on Computationally Efficient VCG-Based Mechanisms for Combinatorial Auctions and Public Projects
https://resolver.caltech.edu/CaltechTHESIS:05242011-112814785
Authors: {'items': [{'email': 'dbuchfuhrer@gmail.com', 'id': 'Buchfuhrer-David-Isaac', 'name': {'family': 'Buchfuhrer', 'given': 'David Isaac'}, 'show_email': 'NO'}]}
Year: 2011
DOI: 10.7907/N0M7-C473
<p>A natural goal in designing mechanisms for auctions and public projects is to maximize the social welfare while incentivizing players to bid truthfully. If these are the only concerns, the problem is easily solved by use of the VCG mechanism. Unfortunately, this mechanism is not computationally efficient in general and there are currently no other general methods for designing truthful mechanisms. However, it is possible to design computationally efficient VCG-based mechanisms which approximately maximize the social welfare.</p>
<p>We explore the design space of computationally efficient VCG-based mechanisms under submodular valuations and show that the achievable approximation guarantees are poor, even compared to efficient non-truthful algorithms. Some of these approximation hardness results stem from an asymmetry in the information available to the players versus that available to the mechanism. We develop an alternative Instance Oracle model which reduces this asymmetry by allowing the mechanism to access some computational capabilities of the players. By building assumptions about player computation into the model, a more realistic study of mechanism design can be undertaken.</p>
<p>Finally, we give VCG-based mechanisms for some problems in the Instance Oracle model which achieve provably better approximations than the best VCG-based mechanism in the standard model. However, for other problems we give reductions in the Instance Oracle model which prove inapproximability results as strong as those shown in the standard model. These provide more robust hardness results that are not simply artifacts of the asymmetry in the standard model.</p>https://thesis.library.caltech.edu/id/eprint/6424A Geometric Analysis of Convex Demixing
https://resolver.caltech.edu/CaltechTHESIS:05202013-091317123
Authors: {'items': [{'email': 'michael.b.mccoy@gmail.com', 'id': 'McCoy-Michael-Brian', 'name': {'family': 'McCoy', 'given': 'Michael Brian'}, 'orcid': '0000-0002-9479-2090', 'show_email': 'NO'}]}
Year: 2013
DOI: 10.7907/156S-EZ89
<p>Demixing is the task of identifying multiple signals given only their sum and prior information about their structures. Examples of demixing problems include (i) separating a signal that is sparse with respect to one basis from a signal that is sparse with respect to a second basis; (ii) decomposing an observed matrix into low-rank and sparse components; and (iii) identifying a binary codeword with impulsive corruptions. This thesis describes and analyzes a convex optimization framework for solving an array of demixing problems.</p>
<p>Our framework includes a random orientation model for the constituent signals that ensures the structures are incoherent. This work introduces a summary parameter, the statistical dimension, that reflects the intrinsic complexity of a signal. The main result indicates that the difficulty of demixing under this random model depends only on the total complexity of the constituent signals involved: demixing succeeds with high probability when the sum of the complexities is less than the ambient dimension; otherwise, it fails with high probability.</p>
<p>The fact that a phase transition between success and failure occurs in demixing is a consequence of a new inequality in conic integral geometry. Roughly speaking, this inequality asserts that a convex cone behaves like a subspace whose dimension is equal to the statistical dimension of the cone. When combined with a geometric optimality condition for demixing, this inequality provides precise quantitative information about the phase transition, including the location and width of the transition region.</p> https://thesis.library.caltech.edu/id/eprint/7726Blackbox Reconstruction of Depth Three Circuits with Top Fan-In Two
https://resolver.caltech.edu/CaltechTHESIS:06082016-032155301
Authors: {'items': [{'email': 'sinhagaur88@gmail.com', 'id': 'Sinha-Gaurav', 'name': {'family': 'Sinha', 'given': 'Gaurav'}, 'orcid': '0000-0002-3590-9543', 'show_email': 'YES'}]}
Year: 2016
DOI: 10.7907/Z92N507D
<p>Reconstruction of arithmetic circuits has been heavily studied in the past few years and has connections to proving lower bounds and deterministic identity testing. In
this thesis we present a polynomial time randomized algorithm for reconstructing ΣΠΣ(2) circuits over characteristic zero fields F i.e. depth−3 circuits with fan-in 2 at the top addition gate and having coefficients from a field of characteristic zero.</p>
<p>The algorithm needs only a black-box query access to the polynomial f ∈ F[x1,...,xn] of degree d, computable by a ΣΠΣ(2) circuit C. In addition, we assume that the
"simple rank" of this polynomial (essential number of variables after removing the g.c.d. of the two multiplication gates) is bigger than a fixed constant. Our algorithm runs in time polynomial in n and d and with high probability returns an equivalent ΣΠΣ(2) circuit.</p>
<p>The problem of reconstructing ΣΠΣ(2) circuits over finite fields was first proposed by Shpilka [27]. The generalization to ΣΠΣ(k) circuits, k = O(1) (over finite
fields) was addressed by Karnin and Shpilka in [18]. The techniques in these previous involve iterating over all objects of certain kinds over the ambient field and thus
the running time depends on the size of the field F. Their reconstruction algorithm uses lower bounds on the lengths of linear locally decodable codes with 2 queries.</p>
<p>In our setting, such ideas immediately pose a problem and we need new techniques.</p>
<p>Our main techniques are based on the use of quantitative Sylvester Gallai theorems from the work of Barak et.al. [3] to find a small collection of "nice" subspaces to
project onto. The heart of this work lies in subtle applications of the quantitative Sylvester Gallai theorems to prove why projections w.r.t. the "nice" subspaces can
be ”glued”. We also use Brill’s equations from [9] to construct a small set of candidate linear forms (containing linear forms from both gates). Another important
technique which comes very handy is the polynomial time randomized algorithm for factoring multivariate polynomials given by Kaltofen [17].</p>https://thesis.library.caltech.edu/id/eprint/9861P-Schemes and Deterministic Polynomial Factoring Over Finite Fields
https://resolver.caltech.edu/CaltechTHESIS:06012017-013622968
Authors: {'items': [{'email': 'magicfoo@gmail.com', 'id': 'Guo-Zeyu', 'name': {'family': 'Guo', 'given': 'Zeyu'}, 'orcid': '0000-0001-7893-4346', 'show_email': 'NO'}]}
Year: 2017
DOI: 10.7907/Z94F1NSG
<p>We introduce a family of mathematical objects called P-schemes, where P is a poset of subgroups of a finite group G. A P-scheme is a collection of partitions of the right coset spaces H\G, indexed by H∈P, that satisfies a list of axioms. These objects generalize the classical notion of association schemes [BI84] as well as the notion of m-schemes [IKS09].</p>
<p>Based on P-schemes, we develop a unifying framework for the problem of deterministic factoring of univariate polynomials over finite field under the generalized Riemann hypothesis (GRH). More specifically, our results include the following:</p>
<p>We show an equivalence between m-scheme as introduced in [IKS09] and P-schemes in the special setting that G is an multiply transitive permutation group and P is a poset of pointwise stabilizers, and therefore realize the theory of m-schemes as part of the richer theory of P-schemes. </p>
<p>We give a generic deterministic algorithm that computes the factorization of the input polynomial ƒ(X) ∈ F<sub>q</sub>[X] given a "lifted polynomial" ƒ~(X) of ƒ(X) and a collection F of "effectively constructible" subfields of the splitting field of ƒ~(X) over a certain base field. It is routine to compute ƒ~(X) from ƒ(X) by lifting the coefficients of ƒ(X) to a number ring. The algorithm then successfully factorizes ƒ(X) under GRH in time polynomial in the size of ƒ~(X) and F, provided that a certain condition concerning P-schemes is satisfied, for P being the poset of subgroups of the Galois group G of ƒ~(X) defined by F via the Galois correspondence. By considering various choices of G, P and verifying the condition, we are able to derive the main results of known (GRH-based) deterministic factoring algorithms [Hua91a; Hua91b; Ron88; Ron92; Evd92; Evd94; IKS09] from our generic algorithm in a uniform way.</p>
<p>We investigate the schemes conjecture in [IKS09] and formulate analogous conjectures associated with various families of permutation groups, each of which has applications on deterministic polynomial factoring. Using a technique called induction of P-schemes, we establish reductions among these conjectures and show that they form a hierarchy of relaxations of the original schemes conjecture.</p>
<p>We connect the complexity of deterministic polynomial factoring with the complexity of the Galois group G of ƒ~(X). Specifically, using techniques from permutation group theory, we obtain a (GRH-based) deterministic factoring algorithm whose running time is bounded in terms of the noncyclic composition factors of G. In particular, this algorithm runs in polynomial time if G is in Γ<sub>k</sub> for some k=2<sup>O(√(log n)</sup>, where Γ<sub>k</sub> denotes the family of finite groups whose noncyclic composition factors are all isomorphic of subgroups of the symmetric group of degree k. Previously, polynomial-time algorithms for Γ<sub>k</sub> were known only for bounded k.</p>
<p>We discuss various aspects of the theory of P-schemes, including techniques of constructing new P-schemes from old ones, P-schemes for symmetric groups and linear groups, orbit P-schemes, etc. For the closely related theory of m-schemes, we provide explicit constructions of strongly antisymmetric homogeneous m-schemes for m≤3. We also show that all antisymmetric homogeneous orbit 3-schemes have a matching for m≥3, improving a result in [IKS09] that confirms the same statement for m≥4.</p>
<p>In summary, our framework reduces the algorithmic problem of deterministic polynomial factoring over finite fields to a combinatorial problem concerning P-schemes, allowing us to not only recover most of the known results but also discover new ones. We believe progress in understanding P-schemes associated with various families of permutation groups will shed some light on the ultimate goal of solving deterministic polynomial factoring over finite fields in polynomial time.</p>https://thesis.library.caltech.edu/id/eprint/10241Combinatorial and Algebraic Propeties of Nonnegative Matrices
https://resolver.caltech.edu/CaltechTHESIS:06062022-043503154
Authors: {'items': [{'email': 'jenishc@gmail.com', 'id': 'Mehta-Jenish-Chetan', 'name': {'family': 'Mehta', 'given': 'Jenish Chetan'}, 'show_email': 'YES'}]}
Year: 2022
DOI: 10.7907/3vxb-6778
<p>We study the combinatorial and algebraic properties of Nonnegative Matrices. Our results are divided into three different categories.</p>
<p>1. We show the first quantitative generalization of the 100 year-old Perron-Frobenius theorem, a fundamental theorem which has been used within diverse areas of mathematics. The Perron-Frobenius theorem shows that any irreducible nonnegative matrix <i>R</i> will have a largest positive eigenvalue <i>r</i>, and every other eigenvalue <i>λ</i> is such that Re<i>λ</i> < <i>R</i> and |λ| ≤ <i>r</i>. We capture the notion of irreducibility through the widely studied notion of edge expansion <i>φ</i> of <i>R</i> which intuitively measures how well-connected the underlying digraph of <i>R</i> is, and show a quantitative relation between the spectral gap Δ = 1-Re<i>λ</i>/<i>r</i> (where <i>λ</i> ≠ <i>r</i> has the largest real part) of <i>R</i> to the edge expansion <i>φ</i> as follows.</p>
<p>(1/15) • [(Δ(<i>R</i>))/n] ≤ <i>φ</i>(<i>R</i>) ≤ √[2 • Δ(<i>R</i>)].</p>
<p>This also provides a more general result than the Cheeger-Buser inequalities since it applies to any nonnegative matrix.</p>
<p>2. We study constructions of specific nonsymmetric matrices (or nonreversible Markov Chains) that have small edge expansion but large spectral gap, taking us in a direction more novel and unexplored than studying symmetric matrices with constant edge expansion that have been extensively studied. We first analyze some known but less studied Markov Chains, and then provide a novel construction of a nonreversible chain for which</p>
<p><i>φ</i>(<i>R</i>) ≤ [(Δ(<i>R</i>))/√<i>n</i>],</p>
<p>obtaining a bound exponentially better than known bounds. We also present a candidate construction of matrices for which</p>
<p><i>φ</i>(<i>R</i>) ≤ 2[(Δ(<i>R</i>))/<i>n</i>]</p>
<p>which is the most beautiful contribution of this thesis. We believe these matrices have properties remarkable enough to deserve study in their own right.</p>
<p>3. We connect the edge expansion and spectral gap to other combinatorial properties of nonsymmetric matrices. The most well-studied property is mixing time, and we provide elementary proofs of the relation between mixing time and the edge expansion, and also other bounds relating the mixing time of a nonreversible chain to the spectral gap and to its additive symmetrization. Further, we provide a unified view of the notion of capacity and normalized capacity, and show the monotonicity of capacity of nonreversible chains amongst other results for nonsymmetric matrices. We finally discuss and prove interesting lemmas about different notions of expansion and show the first results for tensor walks or nonnegative tensors.</p>https://thesis.library.caltech.edu/id/eprint/14949The Identification of Discrete Mixture Models
https://resolver.caltech.edu/CaltechTHESIS:02072023-112938936
Authors: {'items': [{'email': 'sgord512@gmail.com', 'id': 'Gordon-Spencer-Lane', 'name': {'family': 'Gordon', 'given': 'Spencer Lane'}, 'orcid': '0000-0002-7101-2370', 'show_email': 'NO'}]}
Year: 2023
DOI: 10.7907/ebf5-0b48
In this thesis we discuss a variety of results on learning and identifying discrete mixture models, i.e., distributions that are a convex combination of k from a known class C of distributions. We first consider the case where C is the class of binomial distributions, before generalizing to the case of product distributions. We provide a necessary condition for identifiability of mixture of products distributions as well as a generalization to structured mixtures over multiple latent variables.https://thesis.library.caltech.edu/id/eprint/15101