arXiv subjects

Bala Rajaratnam

Publications and source records attributed to Bala Rajaratnam.

At least 37 records · Page 2Linked to original sources

arXiv2016

Model-free consistency of graph partitioning

In this paper, we exploit the theory of dense graph limits to provide a new framework to study the stability of graph partitioning methods, which we call structural consistency. Both stability under perturbation as well as asymptotic consistency (i.e., convergence with probability $1$ as the sample size goes to infinity under a fixed probability model) follow from our notion of structural consistency. By formulating structural consistency as a continuity result on the graphon space, we obtain robust results that are completely independent of the data generating mechanism. In particular, our results apply in settings where observations are not independent, thereby significantly generalizing the common probabilistic approach where data are assumed to be i.i.d. In order to make precise the notion of structural consistency of graph partitioning, we begin by extending the theory of graph limits to include vertex colored graphons. We then define continuous node-level statistics and prove that graph partitioning based on such statistics is consistent. Finally, we derive the structural consistency of commonly used clustering algorithms in a general model-free setting. These include clustering based on local graph statistics such as homomorphism densities, as well as the popular spectral clustering using the normalized Laplacian. We posit that proving the continuity of clustering algorithms in the graph limit topology can stand on its own as a more robust form of model-free consistency. We also believe that the mathematical framework developed in this paper goes beyond the study of clustering algorithms, and will guide the development of similar model-free frameworks to analyze other procedures in the broader mathematical sciences.

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arXiv2016

Towards a sparse, scalable, and stably positive definite (inverse) covariance estimator

High dimensional covariance estimation and graphical models is a contemporary topic in statistics and machine learning having widespread applications. An important line of research in this regard is to shrink the extreme spectrum of the covariance matrix estimators. A separate line of research in the literature has considered sparse inverse covariance estimation which in turn gives rise to graphical models. In practice, however, a sparse covariance or inverse covariance matrix which is simultaneously well-conditioned and at the same time computationally tractable is desired. There has been little research at the confluence of these three topics. In this paper we consider imposing a condition number constraint to various types of losses used in covariance and inverse covariance matrix estimation. When the loss function can be decomposed as a sum of an orthogonally invariant function of the estimate and its inner product with a function of the sample covariance matrix, we show that a solution path algorithm can be derived, involving a series of ordinary differential equations. The path algorithm is attractive because it provides the entire family of estimates for all possible values of the condition number bound, at the same computational cost of a single estimate with a fixed upper bound. An important finding is that the proximal operator for the condition number constraint, which turns out to be very useful in regularizing loss functions that are not orthogonally invariant and may yield non-positive-definite estimates, can be efficiently computed by this path algorithm. As a concrete illustration of its practical importance, we develop an operator-splitting algorithm that imposes a guarantee of well-conditioning as well as positive definiteness to recently proposed convex pseudo-likelihood based graphical model selection methods.

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arXiv2016

Preserving positivity for matrices with sparsity constraints

Functions preserving Loewner positivity when applied entrywise to positive semidefinite matrices have been widely studied in the literature. Following the work of Schoenberg [Duke Math. J. 9], Rudin [Duke Math. J. 26], and others, it is well-known that functions preserving positivity for matrices of all dimensions are absolutely monotonic (i.e., analytic with nonnegative Taylor coefficients). In this paper, we study functions preserving positivity when applied entrywise to sparse matrices, with zeros encoded by a graph $G$ or a family of graphs $G_n$. Our results generalize Schoenberg and Rudin's results to a modern setting, where functions are often applied entrywise to sparse matrices in order to improve their properties (e.g. better conditioning). The only such result known in the literature is for the complete graph $K_2$. We provide the first such characterization result for a large family of non-complete graphs. Specifically, we characterize functions preserving Loewner positivity on matrices with zeros according to a tree. These functions are multiplicatively midpoint-convex and super-additive. Leveraging the underlying sparsity in matrices thus admits the use of functions which are not necessarily analytic nor absolutely monotonic. We further show that analytic functions preserving positivity on matrices with zeros according to trees can contain arbitrarily long sequences of negative coefficients, thus obviating the need for absolute monotonicity in a very strong sense. This result leads to the question of exactly when absolute monotonicity is necessary when preserving positivity for an arbitrary class of graphs. We then provide a stronger condition in terms of the numerical range of all symmetric matrices, such that functions satisfying this condition on matrices with zeros according to any family of graphs with unbounded degrees are necessarily absolutely monotonic.

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arXiv2015

Critical exponents of graphs

The study of entrywise powers of matrices was originated by Loewner in the pursuit of the Bieberbach conjecture. Since the work of FitzGerald and Horn (1977), it is known that $A^{\circ α} := (a_{ij}^α)$ is positive semidefinite for every entrywise nonnegative $n \times n$ positive semidefinite matrix $A = (a_{ij})$ if and only if $α$ is a positive integer or $α\geq n-2$. This surprising result naturally extends the Schur product theorem, and demonstrates the existence of a sharp phase transition in preserving positivity. In this paper, we study when entrywise powers preserve positivity for matrices with structure of zeros encoded by graphs. To each graph is associated an invariant called its "critical exponent", beyond which every power preserves positivity. In our main result, we determine the critical exponents of all chordal/decomposable graphs, and relate them to the geometry of the underlying graphs. We then examine the critical exponent of important families of non-chordal graphs such as cycles and bipartite graphs. Surprisingly, large families of dense graphs have small critical exponents that do not depend on the number of vertices of the graphs.

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arXiv2015

Extracting Common Time Trends from Concurrent Time Series: Maximum Autocorrelation Factors with Application to Tree Ring Time Series Data

Concurrent time series commonly arise in various applications, including when monitoring the environment such as in air quality measurement networks, weather stations, oceanographic buoys, or in paleo form such as lake sediments, tree rings, ice cores, or coral isotopes, with each monitoring or sampling site providing one of the time series. The goal in such applications is to extract a common time trend or signal in the observed data. Other examples where the goal is to extract a common time trend for multiple time series are in stock price time series, neurological time series, and quality control time series. For this purpose we develop properties of MAF [Maximum Autocorrelation Factors] that linearly combines time series in order to maximize the resulting SNR [signal-to-noise-ratio] where there are multiple smooth signals present in the data. Equivalence is established in a regression setting between MAF and CCA [Canonical Correlation Analysis] even though MAF does not require specific signal knowledge as opposed to CCA. We proceed to derive the theoretical properties of MAF and quantify the SNR advantages of MAF in comparison with PCA [Principal Components Analysis], a commonly used method for linearly combining time series, and compare their statistical sample properties. MAF and PCA are then applied to real and simulated data sets to illustrate MAFs efficacy.

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arXiv2015

MCMC-Based Inference in the Era of Big Data: A Fundamental Analysis of the Convergence Complexity of High-Dimensional Chains

Markov chain Monte Carlo (MCMC) lies at the core of modern Bayesian methodology, much of which would be impossible without it. Thus, the convergence properties of MCMCs have received significant attention, and in particular, proving (geometric) ergodicity is of critical interest. Trust in the ability of MCMCs to sample from modern-day high-dimensional posteriors, however, has been limited by a widespread perception that these chains typically experience serious convergence problems. In this paper, we first demonstrate that contemporary methods for obtaining convergence rates have serious limitations when the dimension grows. We then propose a framework for rigorously establishing the convergence behavior of commonly used high-dimensional MCMCs. In particular, we demonstrate theoretically the precise nature and severity of the convergence problems of popular MCMCs when implemented in high dimensions, including phase transitions in the convergence rates in various $n$ and $p$ regimes, and a universality result across an entire spectrum of models. We also show that convergence problems effectively eliminate the apparent safeguard of geometric ergodicity. We then demonstrate theoretical principles by which MCMCs can be constructed and analyzed to yield bounded geometric convergence rates even as the dimension $p$ grows without bound. Additionally, we propose a diagnostic tool for establishing convergence.

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arXiv2015

Integration and measures on the space of countable labelled graphs

In this paper we develop a rigorous foundation for the study of integration and measures on the space $\mathscr{G}(V)$ of all graphs defined on a countable labelled vertex set $V$. We first study several interrelated $σ$-algebras and a large family of probability measures on graph space. We then focus on a "dyadic" Hamming distance function $\left\| \cdot \right\|_{ψ,2}$, which was very useful in the study of differentiation on $\mathscr{G}(V)$. The function $\left\| \cdot \right\|_{ψ,2}$ is shown to be a Haar measure-preserving bijection from the subset of infinite graphs to the circle (with the Haar/Lebesgue measure), thereby naturally identifying the two spaces. As a consequence, we establish a "change of variables" formula that enables the transfer of the Riemann-Lebesgue theory on $\mathbb{R}$ to graph space $\mathscr{G}(V)$. This also complements previous work in which a theory of Newton-Leibnitz differentiation was transferred from the real line to $\mathscr{G}(V)$ for countable $V$. Finally, we identify the Pontryagin dual of $\mathscr{G}(V)$, and characterize the positive definite functions on $\mathscr{G}(V)$.

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arXiv2015

Statistical paleoclimate reconstructions via Markov random fields

Understanding centennial scale climate variability requires data sets that are accurate, long, continuous and of broad spatial coverage. Since instrumental measurements are generally only available after 1850, temperature fields must be reconstructed using paleoclimate archives, known as proxies. Various climate field reconstructions (CFR) methods have been proposed to relate past temperature to such proxy networks. In this work, we propose a new CFR method, called GraphEM, based on Gaussian Markov random fields embedded within an EM algorithm. Gaussian Markov random fields provide a natural and flexible framework for modeling high-dimensional spatial fields. At the same time, they provide the parameter reduction necessary for obtaining precise and well-conditioned estimates of the covariance structure, even in the sample-starved setting common in paleoclimate applications. In this paper, we propose and compare the performance of different methods to estimate the graphical structure of climate fields, and demonstrate how the GraphEM algorithm can be used to reconstruct past climate variations. The performance of GraphEM is compared to the widely used CFR method RegEM with regularization via truncated total least squares, using synthetic data. Our results show that GraphEM can yield significant improvements, with uniform gains over space, and far better risk properties. We demonstrate that the spatial structure of temperature fields can be well estimated by graphs where each neighbor is only connected to a few geographically close neighbors, and that the increase in performance is directly related to recovering the underlying sparsity in the covariance of the spatial field. Our work demonstrates how significant improvements can be made in climate reconstruction methods by better modeling the covariance structure of the climate field.

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arXiv2015

Foundational principles for large scale inference: Illustrations through correlation mining

When can reliable inference be drawn in the "Big Data" context? This paper presents a framework for answering this fundamental question in the context of correlation mining, with implications for general large scale inference. In large scale data applications like genomics, connectomics, and eco-informatics the dataset is often variable-rich but sample-starved: a regime where the number $n$ of acquired samples (statistical replicates) is far fewer than the number $p$ of observed variables (genes, neurons, voxels, or chemical constituents). Much of recent work has focused on understanding the computational complexity of proposed methods for "Big Data." Sample complexity however has received relatively less attention, especially in the setting when the sample size $n$ is fixed, and the dimension $p$ grows without bound. To address this gap, we develop a unified statistical framework that explicitly quantifies the sample complexity of various inferential tasks. Sampling regimes can be divided into several categories: 1) the classical asymptotic regime where the variable dimension is fixed and the sample size goes to infinity; 2) the mixed asymptotic regime where both variable dimension and sample size go to infinity at comparable rates; 3) the purely high dimensional asymptotic regime where the variable dimension goes to infinity and the sample size is fixed. Each regime has its niche but only the latter regime applies to exa-scale data dimension. We illustrate this high dimensional framework for the problem of correlation mining, where it is the matrix of pairwise and partial correlations among the variables that are of interest. We demonstrate various regimes of correlation mining based on the unifying perspective of high dimensional learning rates and sample complexity for different structured covariance models and different inference tasks.

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arXiv2015

Bayesian inference for Gaussian graphical models beyond decomposable graphs

Bayesian inference for graphical models has received much attention in the literature in recent years. It is well known that when the graph G is decomposable, Bayesian inference is significantly more tractable than in the general non-decomposable setting. Penalized likelihood inference on the other hand has made tremendous gains in the past few years in terms of scalability and tractability. Bayesian inference, however, has not had the same level of success, though a scalable Bayesian approach has its respective strengths, especially in terms of quantifying uncertainty. To address this gap, we propose a scalable and flexible novel Bayesian approach for estimation and model selection in Gaussian undirected graphical models. We first develop a class of generalized G-Wishart distributions with multiple shape parameters for an arbitrary underlying graph. This class contains the G-Wishart distribution as a special case. We then introduce the class of Generalized Bartlett (GB) graphs, and derive an efficient Gibbs sampling algorithm to obtain posterior draws from generalized G-Wishart distributions corresponding to a GB graph. The class of Generalized Bartlett graphs contains the class of decomposable graphs as a special case, but is substantially larger than the class of decomposable graphs. We proceed to derive theoretical properties of the proposed Gibbs sampler. We then demonstrate that the proposed Gibbs sampler is scalable to significantly higher dimensional problems as compared to using an accept-reject or a Metropolis-Hasting algorithm. Finally, we show the efficacy of the proposed approach on simulated and real data.

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arXiv2015

Differential Calculus on Graphon Space

Recently, the theory of dense graph limits has received attention from multiple disciplines including graph theory, computer science, statistical physics, probability, statistics, and group theory. In this paper we initiate the study of the general structure of differentiable graphon parameters $F$. We derive consistency conditions among the higher Gâteaux derivatives of $F$ when restricted to the subspace of edge weighted graphs $\mathcal{W}_{\bf p}$. Surprisingly, these constraints are rigid enough to imply that the multilinear functionals $Λ: \mathcal{W}_{\bf p}^n \to \mathbb{R}$ satisfying the constraints are determined by a finite set of constants indexed by isomorphism classes of multigraphs with $n$ edges and no isolated vertices. Using this structure theory, we explain the central role that homomorphism densities play in the analysis of graphons, by way of a new combinatorial interpretation of their derivatives. In particular, homomorphism densities serve as the monomials in a polynomial algebra that can be used to approximate differential graphon parameters as Taylor polynomials. These ideas are summarized by our main theorem, which asserts that homomorphism densities $t(H,-)$ where $H$ has at most $N$ edges form a basis for the space of smooth graphon parameters whose $(N+1)$st derivatives vanish. As a consequence of this theory, we also extend and derive new proofs of linear independence of multigraph homomorphism densities, and characterize homomorphism densities. In addition, we develop a theory of series expansions, including Taylor's theorem for graph parameters and a uniqueness principle for series. We use this theory to analyze questions raised by Lovász, including studying infinite quantum algebras and the connection between right- and left-homomorphism densities.

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arXiv2015

Graphical Markov models for infinitely many variables

Representing the conditional independences present in a multivariate random vector via graphs has found widespread use in applications, and such representations are popularly known as graphical models or Markov random fields. These models have many useful properties, but their fundamental attractive feature is their ability to reflect conditional independences between blocks of variables through graph separation, a consequence of the equivalence of the pairwise, local and global Markov properties demonstrated by Pearl and Paz (1985). Modern day applications often necessitate working with either an infinite collection of variables (such as in a spatial-temporal field) or approximating a large high-dimensional finite stochastic system with an infinite-dimensional system. However, it is unclear whether the conditional independences present in an infinite-dimensional random vector or stochastic process can still be represented by separation criteria in an infinite graph. In light of the advantages of using graphs as tools to represent stochastic relationships, we undertake in this paper a general study of infinite graphical models. First, we demonstrate that naive extensions of the assumptions required for the finite case results do not yield equivalence of the Markov properties in the infinite-dimensional setting, thus calling for a more in-depth analysis. To this end, we proceed to derive general conditions which do allow representing the conditional independence in an infinite-dimensional random system by means of graphs, and our results render the result of Pearl and Paz as a special case of a more general phenomenon. We conclude by demonstrating the applicability of our theory through concrete examples of infinite-dimensional graphical models.

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arXiv2015

Convergence of cyclic coordinatewise l1 minimization

We consider the general problem of minimizing an objective function which is the sum of a convex function (not strictly convex) and absolute values of a subset of variables (or equivalently the l1-norm of the variables). This problem appears exten- sively in modern statistical applications associated with high-dimensional data or "big data", and corresponds to optimizing l1-regularized likelihoods in the context of model selection. In such applications, cyclic coordinatewise minimization (CCM), where the objective function is sequentially minimized with respect to each individual coordi- nate, is often employed as it offers a computationally cheap and effective optimization method. Consequently, it is crucial to obtain theoretical guarantees of convergence for the sequence of iterates produced by the cyclic coordinatewise minimization in this setting. Moreover, as the objective corresponds to at l1-regularized likelihoods of many variables, it is important to obtain convergence of the iterates themselves, and not just the function values. Previous results in the literature only establish either, (i) that every limit point of the sequence of iterates is a stationary point of the objective function, or (ii) establish convergence under special assumptions, or (iii) establish con- vergence for a different minimization approach (which uses quadratic approximation based gradient descent followed by an inexact line search), (iv) establish convergence of only the function values of the sequence of iterates produced by random coordinatewise minimization (a variant of CCM). In this paper, a rigorous general proof of convergence for the cyclic coordinatewise minimization algorithm is provided. We demonstrate the usefulness of our general results in contemporary applications.

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arXiv2015

Lasso Regression: Estimation and Shrinkage via Limit of Gibbs Sampling

The application of the lasso is espoused in high-dimensional settings where only a small number of the regression coefficients are believed to be nonzero. Moreover, statistical properties of high-dimensional lasso estimators are often proved under the assumption that the correlation between the predictors is bounded. In this vein, coordinatewise methods, the most common means of computing the lasso solution, work well in the presence of low to moderate multicollinearity. The computational speed of coordinatewise algorithms degrades however as sparsity decreases and multicollinearity increases. Motivated by these limitations, we propose the novel "Deterministic Bayesian Lasso" algorithm for computing the lasso solution. This algorithm is developed by considering a limiting version of the Bayesian lasso. The performance of the Deterministic Bayesian Lasso improves as sparsity decreases and multicollinearity increases, and can offer substantial increases in computational speed. A rigorous theoretical analysis demonstrates that (1) the Deterministic Bayesian Lasso algorithm converges to the lasso solution, and (2) it leads to a representation of the lasso estimator which shows how it achieves both $\ell_1$ and $\ell_2$ types of shrinkage simultaneously. Connections to other algorithms are also provided. The benefits of the Deterministic Bayesian Lasso algorithm are then illustrated on simulated and real data.

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arXiv2014

Complete characterization of Hadamard powers preserving Loewner positivity, monotonicity, and convexity

Entrywise powers of symmetric matrices preserving positivity, monotonicity or convexity with respect to the Loewner ordering arise in various applications, and have received much attention recently in the literature. Following FitzGerald and Horn [J. Math. Anal. Appl., 1977], it is well-known that there exists a critical exponent beyond which all entrywise powers preserve positive definiteness. Similar phase transition phenomena have also recently been shown by Hiai (2009) to occur for monotonicity and convexity. In this paper, we complete the characterization of all the entrywise powers below and above the critical exponents that are positive, monotone, or convex on the cone of positive semidefinite matrices. We then extend the original problem by fully classifying the positive, monotone, or convex powers in a more general setting where additional rank constraints are imposed on the matrices. We also classify the entrywise powers that are super/sub-additive with respect to the Loewner ordering. Finally, we extend all the previous characterizations to matrices with negative entries. Our analysis consequently allows us to answer a question raised by Bhatia and Elsner (2007) regarding the smallest dimension for which even extensions of the power functions do not preserve Loewner positivity.

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arXiv2014

Optimization Methods for Sparse Pseudo-Likelihood Graphical Model Selection

Sparse high dimensional graphical model selection is a popular topic in contemporary machine learning. To this end, various useful approaches have been proposed in the context of $\ell_1$-penalized estimation in the Gaussian framework. Though many of these inverse covariance estimation approaches are demonstrably scalable and have leveraged recent advances in convex optimization, they still depend on the Gaussian functional form. To address this gap, a convex pseudo-likelihood based partial correlation graph estimation method (CONCORD) has been recently proposed. This method uses coordinate-wise minimization of a regression based pseudo-likelihood, and has been shown to have robust model selection properties in comparison with the Gaussian approach. In direct contrast to the parallel work in the Gaussian setting however, this new convex pseudo-likelihood framework has not leveraged the extensive array of methods that have been proposed in the machine learning literature for convex optimization. In this paper, we address this crucial gap by proposing two proximal gradient methods (CONCORD-ISTA and CONCORD-FISTA) for performing $\ell_1$-regularized inverse covariance matrix estimation in the pseudo-likelihood framework. We present timing comparisons with coordinate-wise minimization and demonstrate that our approach yields tremendous payoffs for $\ell_1$-penalized partial correlation graph estimation outside the Gaussian setting, thus yielding the fastest and most scalable approach for such problems. We undertake a theoretical analysis of our approach and rigorously demonstrate convergence, and also derive rates thereof.

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arXiv2014

A convex pseudo-likelihood framework for high dimensional partial correlation estimation with convergence guarantees

Sparse high dimensional graphical model selection is a topic of much interest in modern day statistics. A popular approach is to apply l1-penalties to either (1) parametric likelihoods, or, (2) regularized regression/pseudo-likelihoods, with the latter having the distinct advantage that they do not explicitly assume Gaussianity. As none of the popular methods proposed for solving pseudo-likelihood based objective functions have provable convergence guarantees, it is not clear if corresponding estimators exist or are even computable, or if they actually yield correct partial correlation graphs. This paper proposes a new pseudo-likelihood based graphical model selection method that aims to overcome some of the shortcomings of current methods, but at the same time retain all their respective strengths. In particular, we introduce a novel framework that leads to a convex formulation of the partial covariance regression graph problem, resulting in an objective function comprised of quadratic forms. The objective is then optimized via a coordinate-wise approach. The specific functional form of the objective function facilitates rigorous convergence analysis leading to convergence guarantees; an important property that cannot be established using standard results, when the dimension is larger than the sample size, as is often the case in high dimensional applications. These convergence guarantees ensure that estimators are well-defined under very general conditions, and are always computable. In addition, the approach yields estimators that have good large sample properties and also respect symmetry. Furthermore, application to simulated/real data, timing comparisons and numerical convergence is demonstrated. We also present a novel unifying framework that places all graphical pseudo-likelihood methods as special cases of a more general formulation, leading to important insights.

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arXiv2014

The Letac-Massam conjecture and existence of high dimensional Bayes estimators for Graphical Models

In recent years, a variety of useful extensions of the Wishart have been proposed in the literature for the purposes of studying Markov random fields/graphical models. In particular, generalizations of the Wishart, referred to as Type I and Type II Wishart distributions, have been introduced by Letac and Massam (\emph{Annals of Statistics} 2006) and play important roles in both frequentist and Bayesian inference for Gaussian graphical models. These distributions have been especially useful in high-dimensional settings due to the flexibility offered by their multiple shape parameters. The domain of In this paper we resolve a long-standing conjecture of Letac and Massam (LM) concerning the domains of the multi-parameters of graphical Wishart type distributions. This conjecture, posed in \emph{Annals of Statistics}, also relates fundamentally to the existence of Bayes estimators corresponding to these high dimensional priors. To achieve our goal, we first develop novel theory in the context of probabilistic analysis of graphical models. Using these tools, and a recently introduced class of Wishart distributions for directed acyclic graph (DAG) models, we proceed to give counterexamples to the LM conjecture, thus completely resolving the problem. Our analysis also proceeds to give useful insights on graphical Wishart distributions with implications for Bayesian inference for such models.

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