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Venkat Chandrasekaran

Publications and source records attributed to Venkat Chandrasekaran.

At least 37 records · Page 2Linked to original sources
arXiv2020

Learning Exponential Family Graphical Models with Latent Variables using Regularized Conditional Likelihood

Fitting a graphical model to a collection of random variables given sample observations is a challenging task if the observed variables are influenced by latent variables, which can induce significant confounding statistical dependencies among the observed variables. We present a new convex relaxation framework based on regularized conditional likelihood for latent-variable graphical modeling in which the conditional distribution of the observed variables conditioned on the latent variables is given by an exponential family graphical model. In comparison to previously proposed tractable methods that proceed by characterizing the marginal distribution of the observed variables, our approach is applicable in a broader range of settings as it does not require knowledge about the specific form of distribution of the latent variables and it can be specialized to yield tractable approaches to problems in which the observed data are not well-modeled as Gaussian. We demonstrate the utility and flexibility of our framework via a series of numerical experiments on synthetic as well as real data.

stat.ML↗
arXiv2020

False Discovery and Its Control in Low Rank Estimation

Models specified by low-rank matrices are ubiquitous in contemporary applications. In many of these problem domains, the row/column space structure of a low-rank matrix carries information about some underlying phenomenon, and it is of interest in inferential settings to evaluate the extent to which the row/column spaces of an estimated low-rank matrix signify discoveries about the phenomenon. However, in contrast to variable selection, we lack a formal framework to assess true/false discoveries in low-rank estimation; in particular, the key source of difficulty is that the standard notion of a discovery is a discrete one that is ill-suited to the smooth structure underlying low-rank matrices. We address this challenge via a geometric reformulation of the concept of a discovery, which then enables a natural definition in the low-rank case. We describe and analyze a generalization of the Stability Selection method of Meinshausen and Bühlmann to control for false discoveries in low-rank estimation, and we demonstrate its utility compared to previous approaches via numerical experiments.

stat.ME↗
arXiv2020

Newton Polytopes and Relative Entropy Optimization

Certifying function nonnegativity is a ubiquitous problem in computational mathematics, with especially notable applications in optimization. We study the question of certifying nonnegativity of signomials based on the recently proposed approach of Sums-of-AM/GM-Exponentials (SAGE) decomposition due to the second author and Shah. The existence of a SAGE decomposition is a sufficient condition for nonnegativity of a signomial, and it can be verified by solving a tractable convex relative entropy program. We present new structural properties of SAGE certificates such as a characterization of the extreme rays of the cones associated to these decompositions as well as an appealing form of sparsity preservation. These lead to a number of important consequences such as conditions under which signomial nonnegativity is equivalent to the existence of a SAGE decomposition; our results represent the broadest-known class of nonconvex signomial optimization problems that can be solved efficiently via convex relaxation. The analysis in this paper proceeds by leveraging the interaction between the convex duality underlying SAGE certificates and the face structure of Newton polytopes. While our primary focus is on signomials, we also discuss how our results provide efficient methods for certifying polynomial nonnegativity, with complexity independent of the degree of a polynomial.

math.OC↗
arXiv2019

A Note on Convex Relaxations for the Inverse Eigenvalue Problem

The affine inverse eigenvalue problem consists of identifying a real symmetric matrix with a prescribed set of eigenvalues in an affine space. Due to its ubiquity in applications, various instances of the problem have been widely studied in the literature. Previous algorithmic solutions were typically nonconvex heuristics and were often developed in a case-by-case manner for specific structured affine spaces. In this short note we describe a general family of convex relaxations for the problem by reformulating it as a question of checking feasibility of a system of polynomial equations, and then leveraging tools from the optimization literature to obtain semidefinite programming relaxations. Our system of polynomial equations may be viewed as a matricial analog of polynomial reformulations of 0/1 combinatorial optimization problems, for which semidefinite relaxations have been extensively investigated. We illustrate numerically the utility of our approach in stylized examples that are drawn from various applications.

math.OC↗
arXiv2019

Signomial and Polynomial Optimization via Relative Entropy and Partial Dualization

We describe a generalization of the Sums-of-AM/GM Exponential (SAGE) relaxation methodology for obtaining bounds on constrained signomial and polynomial optimization problems. Our approach leverages the fact that relative entropy based SAGE certificates conveniently and transparently blend with convex duality, in a manner that Sums-of-Squares certificates do not. This more general approach not only retains key properties of ordinary SAGE relaxations (e.g. sparsity preservation), but also inspires a novel perspective-based method of solution recovery. We illustrate the utility of our methodology with a range of examples from the global optimization literature, along with a publicly available software package.

math.OC↗
arXiv2019

Convex Graph Invariant Relaxations For Graph Edit Distance

The edit distance between two graphs is a widely used measure of similarity that evaluates the smallest number of vertex and edge deletions/insertions required to transform one graph to another. It is NP-hard to compute in general, and a large number of heuristics have been proposed for approximating this quantity. With few exceptions, these methods generally provide upper bounds on the edit distance between two graphs. In this paper, we propose a new family of computationally tractable convex relaxations for obtaining lower bounds on graph edit distance. These relaxations can be tailored to the structural properties of the particular graphs via convex graph invariants. Specific examples that we highlight in this paper include constraints on the graph spectrum as well as (tractable approximations of) the stability number and the maximum-cut values of graphs. We prove under suitable conditions that our relaxations are tight (i.e., exactly compute the graph edit distance) when one of the graphs consists of few eigenvalues. We also validate the utility of our framework on synthetic problems as well as real applications involving molecular structure comparison problems in chemistry.

cs.DS↗
arXiv2017

Efficiently characterizing games consistent with perturbed equilibrium observations

We study the problem of characterizing the set of games that are consistent with observed equilibrium play. 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. We illustrate our approach with numerical simulations.

cs.GT↗
arXiv2016

Interpreting Latent Variables in Factor Models via Convex Optimization

Latent or unobserved phenomena pose a significant difficulty in data analysis as they induce complicated and confounding dependencies among a collection of observed variables. Factor analysis is a prominent multivariate statistical modeling approach that addresses this challenge by identifying the effects of (a small number of) latent variables on a set of observed variables. However, the latent variables in a factor model are purely mathematical objects that are derived from the observed phenomena, and they do not have any interpretation associated to them. A natural approach for attributing semantic information to the latent variables in a factor model is to obtain measurements of some additional plausibly useful covariates that may be related to the original set of observed variables, and to associate these auxiliary covariates to the latent variables. In this paper, we describe a systematic approach for identifying such associations. Our method is based on solving computationally tractable convex optimization problems, and it can be viewed as a generalization of the minimum-trace factor analysis procedure for fitting factor models via convex optimization. We analyze the theoretical consistency of our approach in a high-dimensional setting as well as its utility in practice via experimental demonstrations with real data.

stat.ME↗
arXiv2016

California Reservoir Drought Sensitivity and Exhaustion Risk Using Statistical Graphical Models

The ongoing California drought has highlighted the potential vulnerability of state water management infrastructure to multi-year dry intervals. Due to the high complexity of the network, dynamic storage changes across the California reservoir system have been difficult to model using either conventional statistical or physical approaches. Here, we analyze the interactions of monthly volumes in a network of 55 large California reservoirs, over a period of 136 months from 2004 to 2015, and we develop a latent-variable graphical model of their joint fluctuations. We achieve reliable and tractable modeling of the system because the model structure allows unique recovery of the best-in-class model via convex optimization with control of the number of free parameters. We extract a statewide `latent' influencing factor which turns out to be highly correlated with both the Palmer Drought Severity Index (PDSI, $ρ\approx 0.86$) and hydroelectric production ($ρ\approx 0.71$). Further, the model allows us to determine system health measures such as exhaustion probability and per-reservoir drought sensitivity. We find that as PDSI approaches -6, there is a probability greater than 50\% of simultaneous exhaustion of multiple large reservoirs.

stat.AP↗
arXiv2016

Finding Planted Subgraphs with Few Eigenvalues using the Schur-Horn Relaxation

Extracting structured subgraphs inside large graphs - often known as the planted subgraph problem - is a fundamental question that arises in a range of application domains. This problem is NP-hard in general, and as a result, significant efforts have been directed towards the development of tractable procedures that succeed on specific families of problem instances. We propose a new computationally efficient convex relaxation for solving the planted subgraph problem; our approach is based on tractable semidefinite descriptions of majorization inequalities on the spectrum of a symmetric matrix. This procedure is effective at finding planted subgraphs that consist of few distinct eigenvalues, and it generalizes previous convex relaxation techniques for finding planted cliques. Our analysis relies prominently on the notion of spectrally comonotone matrices, which are pairs of symmetric matrices that can be transformed to diagonal matrices with sorted diagonal entries upon conjugation by the same orthogonal matrix.

math.OC↗
arXiv2015

Sufficient Dimension Reduction and Modeling Responses Conditioned on Covariates: An Integrated Approach via Convex Optimization

Given observations of a collection of covariates and responses $(Y, X) \in \mathbb{R}^p \times \mathbb{R}^q$, sufficient dimension reduction (SDR) techniques aim to identify a mapping $f: \mathbb{R}^q \rightarrow \mathbb{R}^k$ with $k \ll q$ such that $Y|f(X)$ is independent of $X$. The image $f(X)$ summarizes the relevant information in a potentially large number of covariates $X$ that influence the responses $Y$. In many contemporary settings, the number of responses $p$ is also quite large, in addition to a large number $q$ of covariates. This leads to the challenge of fitting a succinctly parameterized statistical model to $Y|f(X)$, which is a problem that is usually not addressed in a traditional SDR framework. In this paper, we present a computationally tractable convex relaxation based estimator for simultaneously (a) identifying a linear dimension reduction $f(X)$ of the covariates that is sufficient with respect to the responses, and (b) fitting several types of structured low-dimensional models -- factor models, graphical models, latent-variable graphical models -- to the conditional distribution of $Y|f(X)$. We analyze the consistency properties of our estimator in a high-dimensional scaling regime. We also illustrate the performance of our approach on a newsgroup dataset and on a dataset consisting of financial asset prices.

stat.ME↗
arXiv2015

Regularization for Design

When designing controllers for large-scale systems, the architectural aspects of the controller such as the placement of actuators, sensors, and the communication links between them can no longer be taken as given. The task of designing this architecture is now as important as the design of the control laws themselves. By interpreting controller synthesis (in a model matching setup) as the solution of a particular linear inverse problem, we view the challenge of obtaining a controller with a desired architecture as one of finding a structured solution to an inverse problem. Building on this conceptual connection, we formulate and analyze a framework called \textit{Regularization for Design (RFD)}, in which we augment the variational formulations of controller synthesis problems with convex penalty functions that induce a desired controller architecture. The resulting regularized formulations are convex optimization problems that can be solved efficiently, these convex programs provide a unified computationally tractable approach for the simultaneous co-design of a structured optimal controller and the actuation, sensing and communication architecture required to implement it. Further, these problems are natural control-theoretic analogs of prominent approaches such as the Lasso, the Group Lasso, the Elastic Net, and others that are employed in statistical modeling. In analogy to that literature, we show that our approach identifies optimally structured controllers under a suitable condition on a "signal-to-noise" type ratio.

math.OC↗
arXiv2015

High-Dimensional Change-Point Estimation: Combining Filtering with Convex Optimization

We consider change-point estimation in a sequence of high-dimensional signals given noisy observations. Classical approaches to this problem such as the filtered derivative method are useful for sequences of scalar-valued signals, but they have undesirable scaling behavior in the high-dimensional setting. However, many high-dimensional signals encountered in practice frequently possess latent low-dimensional structure. Motivated by this observation, we propose a technique for high-dimensional change-point estimation that combines the filtered derivative approach from previous work with convex optimization methods based on atomic norm regularization, which are useful for exploiting structure in high-dimensional data. Our algorithm is applicable in online settings as it operates on small portions of the sequence of observations at a time, and it is well-suited to the high-dimensional setting both in terms of computational scalability and of statistical efficiency. The main result of this paper shows that our method performs change-point estimation reliably as long as the product of the smallest-sized change (the Euclidean-norm-squared of the difference between signals at a change-point) and the smallest distance between change-points (number of time instances) is larger than a Gaussian width parameter that characterizes the low-dimensional complexity of the underlying signal sequence.

math.ST↗
arXiv2014

Resource Allocation for Statistical Estimation

Statistical estimation in many contemporary settings involves the acquisition, analysis, and aggregation of datasets from multiple sources, which can have significant differences in character and in value. Due to these variations, the effectiveness of employing a given resource (e.g., a sensing device or computing power) for gathering or processing data from a particular source depends on the nature of that source. As a result, the appropriate division and assignment of a collection of resources to a set of data sources can substantially impact the overall performance of an inferential strategy. In this expository article, we adopt a general view of the notion of a resource and its effect on the quality of a data source, and we describe a framework for the allocation of a given set of resources to a collection of sources in order to optimize a specified metric of statistical efficiency. We discuss several stylized examples involving inferential tasks such as parameter estimation and hypothesis testing based on heterogeneous data sources, in which optimal allocations can be computed either in closed form or via efficient numerical procedures based on convex optimization.

stat.AP↗
arXiv2014

Relative Entropy Relaxations for Signomial Optimization

Signomial programs (SPs) are optimization problems specified in terms of signomials, which are weighted sums of exponentials composed with linear functionals of a decision variable. SPs are non-convex optimization problems in general, and families of NP-hard problems can be reduced to SPs. In this paper we describe a hierarchy of convex relaxations to obtain successively tighter lower bounds of the optimal value of SPs. This sequence of lower bounds is computed by solving increasingly larger-sized relative entropy optimization problems, which are convex programs specified in terms of linear and relative entropy functions. Our approach relies crucially on the observation that the relative entropy function -- by virtue of its joint convexity with respect to both arguments -- provides a convex parametrization of certain sets of globally nonnegative signomials with efficiently computable nonnegativity certificates via the arithmetic-geometric-mean inequality. By appealing to representation theorems from real algebraic geometry, we show that our sequences of lower bounds converge to the global optima for broad classes of SPs. Finally, we also demonstrate the effectiveness of our methods via numerical experiments.

math.OC↗
arXiv2013

Conic Geometric Programming

We introduce and study conic geometric programs (CGPs), which are convex optimization problems that unify geometric programs (GPs) and conic optimization problems such as semidefinite programs (SDPs). A CGP consists of a linear objective function that is to be minimized subject to affine constraints, convex conic constraints, and upper bound constraints on sums of exponential and affine functions. The conic constraints are the central feature of conic programs such as SDPs, while upper bounds on combined exponential/affine functions are generalizations of the types of constraints found in GPs. The dual of a CGP involves the maximization of the negative relative entropy between two nonnegative vectors jointly, subject to affine and conic constraints on the two vectors. Although CGPs contain GPs and SDPs as special instances, computing global optima of CGPs is not much harder than solving GPs and SDPs. More broadly, the CGP framework facilitates a range of new applications that fall outside the scope of SDPs and GPs. Specifically, we demonstrate the utility of CGPs in providing solutions to problems such as permanent maximization, hitting-time estimation in dynamical systems, the computation of the capacity of channels transmitting quantum information, and robust optimization formulations of GPs.

math.OC↗
arXiv2012

Computational and Statistical Tradeoffs via Convex Relaxation

In modern data analysis, one is frequently faced with statistical inference problems involving massive datasets. Processing such large datasets is usually viewed as a substantial computational challenge. However, if data are a statistician's main resource then access to more data should be viewed as an asset rather than as a burden. In this paper we describe a computational framework based on convex relaxation to reduce the computational complexity of an inference procedure when one has access to increasingly larger datasets. Convex relaxation techniques have been widely used in theoretical computer science as they give tractable approximation algorithms to many computationally intractable tasks. We demonstrate the efficacy of this methodology in statistical estimation in providing concrete time-data tradeoffs in a class of denoising problems. Thus, convex relaxation offers a principled approach to exploit the statistical gains from larger datasets to reduce the runtime of inference algorithms.

math.ST↗