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A Caltech Library Repository Feedhttp://www.rssboard.org/rss-specificationpython-feedgenenMon, 15 Apr 2024 15:06:21 +0000Efficiently Characterizing Games Consistent with Perturbed Equilibrium Observations
https://resolver.caltech.edu/CaltechTHESIS:12122016-183248666
Authors: {'items': [{'email': 'juba.ziani@gmail.com', 'id': 'Ziani-Juba', 'name': {'family': 'Ziani', 'given': 'Juba'}, 'orcid': '0000-0002-3324-4349', 'show_email': 'NO'}]}
Year: 2017
DOI: 10.7907/Z91Z42CF
<p>In this thesis, we study the problem of characterizing the set of games that are consistent with observed equilibrium play, a fundamental problem in econometrics. Our contribution is to develop and analyze a new methodology based on convex optimization to address this problem, for many classes of games and observation models of interest. Our approach provides a sharp, computationally efficient characterization of the extent to which a particular set of observations constrains the space of games that could have generated them. This allows us to solve a number of variants of this problem as well as to quantify the power of games from particular classes (e.g., zero-sum, potential, linearly parameterized) to explain player behavior.</p>
<p>We illustrate our approach with numerical simulations.</p>https://thesis.library.caltech.edu/id/eprint/9993Fitting Convex Sets to Data: Algorithms and Applications
https://resolver.caltech.edu/CaltechTHESIS:09282018-091842941
Authors: {'items': [{'email': 'sohyongsheng87@gmail.com', 'id': 'Soh-Yong-Sheng', 'name': {'family': 'Soh', 'given': 'Yong Sheng'}, 'orcid': '0000-0003-3367-1401', 'show_email': 'YES'}]}
Year: 2019
DOI: 10.7907/jkmq-b430
<p>This thesis concerns the geometric problem of finding a convex set that best fits a given dataset. Our question serves as an abstraction for data-analytical tasks arising in a range of scientific and engineering applications. We focus on two specific instances:</p>
<p>1. A key challenge that arises in solving inverse problems is ill-posedness due to a lack of measurements. A prominent family of methods for addressing such issues is based on augmenting optimization-based approaches with a convex penalty function so as to induce a desired structure in the solution. These functions are typically chosen using prior knowledge about the data. In Chapter 2, we study the problem of learning convex penalty functions directly from data for settings in which we lack the domain expertise to choose a penalty function. Our solution relies on suitably transforming the problem of learning a penalty function into a fitting task.</p>
<p>2. In Chapter 3, we study the problem of fitting tractably-described convex sets given the optimal value of linear functionals evaluated in different directions.</p>
<p>Our computational procedures for fitting convex sets are based on a broader framework in which we search among families of sets that are parameterized as linear projections of a fixed structured convex set. The utility of such a framework is that our procedures reduce to the computation of simple primitives at each iteration, and these primitives can be further performed in parallel. In addition, by choosing structured sets that are non-polyhedral, our framework provides a principled way to search over expressive collections of non-polyhedral descriptions; in particular, convex sets that can be described via semidefinite programming provide a rich source of non-polyhedral sets, and such sets feature prominently in this thesis.</p>
<p>We provide performance guarantees for our procedures. Our analyses rely on understanding geometrical aspects of determinantal varieties, building on ideas from empirical processes as well as random matrix theory. We demonstrate the utility of our framework with numerical experiments on synthetic data as well as applications in image denoising and computational geometry.</p>
<p>As secondary contributions, we consider the following:</p>
<p>1. In Chapter 4, we consider the problem of optimally approximating a convex set as a spectrahedron of a given size. Spectrahedra are sets that can be expressed as feasible regions of a semidefinite program.</p>
<p>2. In Chapter 5, we consider change-point estimation in a sequence of high-dimensional signals given noisy observations. Our method integrates classical approaches with a convex optimization-based step that is useful for exploiting structure in high-dimensional data.</p>https://thesis.library.caltech.edu/id/eprint/11208Convex Relaxations for Graph and Inverse Eigenvalue Problems
https://resolver.caltech.edu/CaltechTHESIS:01152020-210801253
Authors: {'items': [{'email': 'utkancandogan@gmail.com', 'id': 'Candogan-Utkan-Onur', 'name': {'family': 'Candogan', 'given': 'Utkan Onur'}, 'orcid': '0000-0002-1416-4909', 'show_email': 'NO'}]}
Year: 2020
DOI: 10.7907/ZV0D-SW58
<p>This thesis is concerned with presenting convex optimization based tractable solutions for three fundamental problems:</p>
<p>1. <i>Planted subgraph problem</i>: Given two graphs, identifying the subset of vertices of the larger graph corresponding to the smaller one.</p>
<p>2. <i>Graph edit distance problem</i>: Given two graphs, calculating the number of edge/vertex additions and deletions required to transform one graph into the other.</p>
<p>3. <i>Affine inverse eigenvalue problem</i>: Given a subspace <b>ε</b> ⊂ 𝕊ⁿ and a vector of eigenvalues λ ∈ ℝⁿ, finding a symmetric matrix with spectrum λ contained in <b>ε</b>.</p>
<p>These combinatorial and algebraic problems frequently arise in various application domains such as social networks, computational biology, chemoinformatics, and control theory. Nevertheless, exactly solving them in practice is only possible for very small instances due to their complexity. For each of these problems, we introduce convex relaxations which succeed in providing exact or approximate solutions in a computationally tractable manner.</p>
<p>Our relaxations for the two graph problems are based on convex graph invariants, which are functions of graphs that do not depend on a particular labeling. One of these convex relaxations, coined the Schur-Horn orbitope, corresponds to the convex hull of all matrices with a given spectrum, and plays a prominent role in this thesis. Specifically, we utilize relaxations based on the Schur-Horn orbitope in the context of the planted subgraph problem and the graph edit distance problem. For both of these problems, we identify conditions under which the Schur-Horn orbitope based relaxations exactly solve the corresponding problem with overwhelming probability. Specifically, we demonstrate that these relaxations turn out to be particularly effective when the underlying graph has a spectrum comprised of few distinct eigenvalues with high multiplicities. In addition to relaxations based on the Schur-Horn orbitope, we also consider outer-approximations based on other convex graph invariants such as the stability number and the maximum-cut value for the graph edit distance problem. On the other hand, for the inverse eigenvalue problem, we investigate two relaxations arising from a sum of squares hierarchy. These relaxations have different approximation qualities, and accordingly induce different computational costs. We utilize our framework to generate solutions for, or certify unsolvability of the underlying inverse eigenvalue problem.</p>
<p>We particularly emphasize the computational aspect of our relaxations throughout this thesis. We corroborate the utility of our methods with various numerical experiments.</p>https://thesis.library.caltech.edu/id/eprint/13622Latent-Variable Modeling: Algorithms, Inference, and Applications
https://resolver.caltech.edu/CaltechTHESIS:09222019-132051506
Authors: {'items': [{'email': '\u200b armeen.taeb@gmail.com', 'id': 'Taeb-Armeen', 'name': {'family': 'Taeb', 'given': 'Armeen'}, 'orcid': '0000-0002-5647-3160', 'show_email': 'YES'}]}
Year: 2020
DOI: 10.7907/YRF1-7W29
<p>Many driving factors of physical systems are often latent or unobserved. Thus, understanding such systems crucially relies on accounting for the influence of the latent structure. This thesis makes advances in three aspects of latent-variable modeling: inference, algorithms, and applications. Specifically, we develop and explore latent-variable techniques that a) ensure interpretable and statistically significant models, b) can be efficiently optimized to identify best fit to data, and c) provide useful insights in real-world applications. The specific contributions of this thesis are:</p>
<p>1. We employ a latent-variable graphical modeling technique to develop the first state-wide statistical model of the California reservoir network. With this model, we precisely characterize the system-wide behavior of the network to hypothetical drought conditions, and proposed guidelines for more sustainable reservoir management.</p>
<p>2. Motivated by the previous application, we provide a geometric framework to assess the extent to which our latent variable model has learned true or false discoveries about the relevant physical phenomena. Our approach generalizes the classical notions of true and false discoveries in mathematical statistics that rely on the discrete structure of the decision space to settings where the decision space is continuous and more complicated. We highlight the utility of this viewpoint in problems involving subspace selection and low-rank estimation.</p>
<p>3. We propose a convex optimization procedure to fit a latent-variable graphical model for generalized linear models. This framework provides a flexible approach to model non-Gaussian variables including Poisson, Bernoulli, and exponential variables. A particularly novel aspect of our formulation is that it incorporates regularizers that are tailored to the type of latent variables.</p>
<p>4. We describe a computationally efficient framework to learn a latent-variable model with high-dimensional and non-iid data. This framework is based on factoriable precision operators that decouple the component associated with the observational dependencies and the component associated to interdependencies among the variables.</p>
<p>5. We propose a convex optimization technique to provide semantics to latent variables of a factor model. This approach is based on linking auxiliary variables -- chosen based on domain expertise -- to these latent variables.</p>https://thesis.library.caltech.edu/id/eprint/11799Applications of Convex Analysis to Signomial and Polynomial Nonnegativity Problems
https://resolver.caltech.edu/CaltechTHESIS:05202021-194439071
Authors: {'items': [{'email': 'rjmurray201693@gmail.com', 'id': 'Murray-Riley-John', 'name': {'family': 'Murray', 'given': 'Riley John'}, 'orcid': '0000-0003-1461-6458', 'show_email': 'NO'}]}
Year: 2021
DOI: 10.7907/vn9x-xj10
<p>Here is a question that is easy to state, but often hard to answer:</p>
<p><i>Is this function nonnegative on this set?</i></p>
<p>When faced with such a question, one often makes appeals to known inequalities. One crafts arguments that are <i>sufficient</i> to establish the nonnegativity of the function, rather than determining the function's precise range of values. This thesis studies sufficient conditions for nonnegativity of signomials and polynomials. Conceptually, signomials may be viewed as generalized polynomials that feature arbitrary real exponents, but with variables restricted to the positive orthant.</p>
<p>Our methods leverage efficient algorithms for a type of convex optimization known as relative entropy programming (REP). By virtue of this integration with REP, our methods can help answer questions like the following:</p>
<p>Is there some function, in this particular space of functions, that is nonnegative on this set?</p>
<p>The ability to answer such questions is <i>extremely</i> useful in applied mathematics.
Alternative approaches in this same vein (e.g., methods for polynomials based on semidefinite programming)
have been used successfully as convex relaxation frameworks for nonconvex optimization, as mechanisms for analyzing dynamical systems, and even as tools for solving nonlinear partial differential equations.</p>
<p>This thesis builds from the <i>sums of arithmetic-geometric exponentials</i> or <i>SAGE</i> approach to signomial nonnegativity. The term "exponential" appears in the SAGE acronym because SAGE parameterizes signomials in terms of exponential functions.</p>
<p>Our first round of contributions concern the original SAGE approach. We employ basic techniques in convex analysis and convex geometry to derive structural results for spaces of SAGE signomials and exactness results for SAGE-based REP relaxations of nonconvex signomial optimization problems.
We frame our analysis primarily in terms of the coefficients of a signomial's basis expansion rather than in terms of signomials themselves.
The effect of this framing is that our results for signomials readily transfer to polynomials. In particular, we are led to define a new concept of <i>SAGE polynomials</i>. For sparse polynomials, this method offers an exponential efficiency improvement relative to certificates of nonnegativity obtained through semidefinite programming.</p>
<p>We go on to create the <i>conditional SAGE</i> methodology for exploiting convex substructure in constrained signomial nonnegativity problems.
The basic insight here is that since the standard relative entropy representation of SAGE signomials is obtained by a suitable application of convex duality, we are free to add additional convex constraints into the duality argument. In the course of explaining this idea we provide some illustrative examples in signomial optimization and analysis of chemical dynamics.</p>
<p>The majority of this thesis is dedicated to exploring fundamental questions surrounding conditional SAGE signomials. We approach these questions through analysis frameworks of <i>sublinear circuits</i> and <i>signomial rings</i>. These sublinear circuits generalize simplicial circuits of affine-linear matroids, and lead to rich modes of analysis for sets that are simultaneously convex in the usual sense and convex under a logarithmic transformation. The concept of signomial rings lets us develop a powerful signomial Positivstellensatz and an elementary signomial moment theory. The Positivstellensatz provides for an effective hierarchy of REP relaxations for approaching the value of a nonconvex signomial minimization problem from below, as well as a first-of-its-kind hierarchy for approaching the same value from above.</p>
<p>In parallel with our mathematical work, we have developed the sageopt python package. Sageopt drives all the examples and experiments used throughout this thesis, and has been used by engineers to solve high-degree polynomial optimization problems at scales unattainable by alternative methods.
We conclude this thesis with an explanation of how our theoretical results affected sageopt's design.</p>https://thesis.library.caltech.edu/id/eprint/14169