arXiv subjects

Pradeep Ravikumar

Publications and source records attributed to Pradeep Ravikumar.

At least 109 records · Page 6Linked to original sources

arXiv2016

Square Root Graphical Models: Multivariate Generalizations of Univariate Exponential Families that Permit Positive Dependencies

We develop Square Root Graphical Models (SQR), a novel class of parametric graphical models that provides multivariate generalizations of univariate exponential family distributions. Previous multivariate graphical models [Yang et al. 2015] did not allow positive dependencies for the exponential and Poisson generalizations. However, in many real-world datasets, variables clearly have positive dependencies. For example, the airport delay time in New York---modeled as an exponential distribution---is positively related to the delay time in Boston. With this motivation, we give an example of our model class derived from the univariate exponential distribution that allows for almost arbitrary positive and negative dependencies with only a mild condition on the parameter matrix---a condition akin to the positive definiteness of the Gaussian covariance matrix. Our Poisson generalization allows for both positive and negative dependencies without any constraints on the parameter values. We also develop parameter estimation methods using node-wise regressions with $\ell_1$ regularization and likelihood approximation methods using sampling. Finally, we demonstrate our exponential generalization on a synthetic dataset and a real-world dataset of airport delay times.

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arXiv2016

Generalized Root Models: Beyond Pairwise Graphical Models for Univariate Exponential Families

We present a novel k-way high-dimensional graphical model called the Generalized Root Model (GRM) that explicitly models dependencies between variable sets of size k > 2---where k = 2 is the standard pairwise graphical model. This model is based on taking the k-th root of the original sufficient statistics of any univariate exponential family with positive sufficient statistics, including the Poisson and exponential distributions. As in the recent work with square root graphical (SQR) models [Inouye et al. 2016]---which was restricted to pairwise dependencies---we give the conditions of the parameters that are needed for normalization using the radial conditionals similar to the pairwise case [Inouye et al. 2016]. In particular, we show that the Poisson GRM has no restrictions on the parameters and the exponential GRM only has a restriction akin to negative definiteness. We develop a simple but general learning algorithm based on L1-regularized node-wise regressions. We also present a general way of numerically approximating the log partition function and associated derivatives of the GRM univariate node conditionals---in contrast to [Inouye et al. 2016], which only provided algorithm for estimating the exponential SQR. To illustrate GRM, we model word counts with a Poisson GRM and show the associated k-sized variable sets. We finish by discussing methods for reducing the parameter space in various situations.

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arXiv2015

Exponential Family Matrix Completion under Structural Constraints

We consider the matrix completion problem of recovering a structured matrix from noisy and partial measurements. Recent works have proposed tractable estimators with strong statistical guarantees for the case where the underlying matrix is low--rank, and the measurements consist of a subset, either of the exact individual entries, or of the entries perturbed by additive Gaussian noise, which is thus implicitly suited for thin--tailed continuous data. Arguably, common applications of matrix completion require estimators for (a) heterogeneous data--types, such as skewed--continuous, count, binary, etc., (b) for heterogeneous noise models (beyond Gaussian), which capture varied uncertainty in the measurements, and (c) heterogeneous structural constraints beyond low--rank, such as block--sparsity, or a superposition structure of low--rank plus elementwise sparseness, among others. In this paper, we provide a vastly unified framework for generalized matrix completion by considering a matrix completion setting wherein the matrix entries are sampled from any member of the rich family of exponential family distributions; and impose general structural constraints on the underlying matrix, as captured by a general regularizer $\mathcal{R}(.)$. We propose a simple convex regularized $M$--estimator for the generalized framework, and provide a unified and novel statistical analysis for this general class of estimators. We finally corroborate our theoretical results on simulated datasets.

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arXiv2015

On Graphical Models via Univariate Exponential Family Distributions

Undirected graphical models, or Markov networks, are a popular class of statistical models, used in a wide variety of applications. Popular instances of this class include Gaussian graphical models and Ising models. In many settings, however, it might not be clear which subclass of graphical models to use, particularly for non-Gaussian and non-categorical data. In this paper, we consider a general sub-class of graphical models where the node-wise conditional distributions arise from exponential families. This allows us to derive multivariate graphical model distributions from univariate exponential family distributions, such as the Poisson, negative binomial, and exponential distributions. Our key contributions include a class of M-estimators to fit these graphical model distributions; and rigorous statistical analysis showing that these M-estimators recover the true graphical model structure exactly, with high probability. We provide examples of genomic and proteomic networks learned via instances of our class of graphical models derived from Poisson and exponential distributions.

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arXiv2015

Vector-Space Markov Random Fields via Exponential Families

We present Vector-Space Markov Random Fields (VS-MRFs), a novel class of undirected graphical models where each variable can belong to an arbitrary vector space. VS-MRFs generalize a recent line of work on scalar-valued, uni-parameter exponential family and mixed graphical models, thereby greatly broadening the class of exponential families available (e.g., allowing multinomial and Dirichlet distributions). Specifically, VS-MRFs are the joint graphical model distributions where the node-conditional distributions belong to generic exponential families with general vector space domains. We also present a sparsistent $M$-estimator for learning our class of MRFs that recovers the correct set of edges with high probability. We validate our approach via a set of synthetic data experiments as well as a real-world case study of over four million foods from the popular diet tracking app MyFitnessPal. Our results demonstrate that our algorithm performs well empirically and that VS-MRFs are capable of capturing and highlighting interesting structure in complex, real-world data. All code for our algorithm is open source and publicly available.

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arXiv2015

Optimal Decision-Theoretic Classification Using Non-Decomposable Performance Metrics

We provide a general theoretical analysis of expected out-of-sample utility, also referred to as decision-theoretic classification, for non-decomposable binary classification metrics such as F-measure and Jaccard coefficient. Our key result is that the expected out-of-sample utility for many performance metrics is provably optimized by a classifier which is equivalent to a signed thresholding of the conditional probability of the positive class. Our analysis bridges a gap in the literature on binary classification, revealed in light of recent results for non-decomposable metrics in population utility maximization style classification. Our results identify checkable properties of a performance metric which are sufficient to guarantee a probability ranking principle. We propose consistent estimators for optimal expected out-of-sample classification. As a consequence of the probability ranking principle, computational requirements can be reduced from exponential to cubic complexity in the general case, and further reduced to quadratic complexity in special cases. We provide empirical results on simulated and benchmark datasets evaluating the performance of the proposed algorithms for decision-theoretic classification and comparing them to baseline and state-of-the-art methods in population utility maximization for non-decomposable metrics.

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arXiv2015

Proximal Quasi-Newton for Computationally Intensive L1-regularized M-estimators

We consider the class of optimization problems arising from computationally intensive L1-regularized M-estimators, where the function or gradient values are very expensive to compute. A particular instance of interest is the L1-regularized MLE for learning Conditional Random Fields (CRFs), which are a popular class of statistical models for varied structured prediction problems such as sequence labeling, alignment, and classification with label taxonomy. L1-regularized MLEs for CRFs are particularly expensive to optimize since computing the gradient values requires an expensive inference step. In this work, we propose the use of a carefully constructed proximal quasi-Newton algorithm for such computationally intensive M-estimation problems, where we employ an aggressive active set selection technique. In a key contribution of the paper, we show that the proximal quasi-Newton method is provably super-linearly convergent, even in the absence of strong convexity, by leveraging a restricted variant of strong convexity. In our experiments, the proposed algorithm converges considerably faster than current state-of-the-art on the problems of sequence labeling and hierarchical classification.

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arXiv2014

On the Information Theoretic Limits of Learning Ising Models

We provide a general framework for computing lower-bounds on the sample complexity of recovering the underlying graphs of Ising models, given i.i.d samples. While there have been recent results for specific graph classes, these involve fairly extensive technical arguments that are specialized to each specific graph class. In contrast, we isolate two key graph-structural ingredients that can then be used to specify sample complexity lower-bounds. Presence of these structural properties makes the graph class hard to learn. We derive corollaries of our main result that not only recover existing recent results, but also provide lower bounds for novel graph classes not considered previously. We also extend our framework to the random graph setting and derive corollaries for Erdős-Rényi graphs in a certain dense setting.

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arXiv2014

A General Framework for Mixed Graphical Models

"Mixed Data" comprising a large number of heterogeneous variables (e.g. count, binary, continuous, skewed continuous, among other data types) are prevalent in varied areas such as genomics and proteomics, imaging genetics, national security, social networking, and Internet advertising. There have been limited efforts at statistically modeling such mixed data jointly, in part because of the lack of computationally amenable multivariate distributions that can capture direct dependencies between such mixed variables of different types. In this paper, we address this by introducing a novel class of Block Directed Markov Random Fields (BDMRFs). Using the basic building block of node-conditional univariate exponential families from Yang et al. (2012), we introduce a class of mixed conditional random field distributions, that are then chained according to a block-directed acyclic graph to form our class of Block Directed Markov Random Fields (BDMRFs). The Markov independence graph structure underlying a BDMRF thus has both directed and undirected edges. We introduce conditions under which these distributions exist and are normalizable, study several instances of our models, and propose scalable penalized conditional likelihood estimators with statistical guarantees for recovering the underlying network structure. Simulations as well as an application to learning mixed genomic networks from next generation sequencing expression data and mutation data demonstrate the versatility of our methods.

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arXiv2014

Sparsistency of $\ell_1$-Regularized $M$-Estimators

We consider the model selection consistency or sparsistency of a broad set of $\ell_1$-regularized $M$-estimators for linear and non-linear statistical models in a unified fashion. For this purpose, we propose the local structured smoothness condition (LSSC) on the loss function. We provide a general result giving deterministic sufficient conditions for sparsistency in terms of the regularization parameter, ambient dimension, sparsity level, and number of measurements. We show that several important statistical models have $M$-estimators that indeed satisfy the LSSC, and as a result, the sparsistency guarantees for the corresponding $\ell_1$-regularized $M$-estimators can be derived as simple applications of our main theorem.

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arXiv2013

Sparse Inverse Covariance Matrix Estimation Using Quadratic Approximation

The L1-regularized Gaussian maximum likelihood estimator (MLE) has been shown to have strong statistical guarantees in recovering a sparse inverse covariance matrix, or alternatively the underlying graph structure of a Gaussian Markov Random Field, from very limited samples. We propose a novel algorithm for solving the resulting optimization problem which is a regularized log-determinant program. In contrast to recent state-of-the-art methods that largely use first order gradient information, our algorithm is based on Newton's method and employs a quadratic approximation, but with some modifications that leverage the structure of the sparse Gaussian MLE problem. We show that our method is superlinearly convergent, and present experimental results using synthetic and real-world application data that demonstrate the considerable improvements in performance of our method when compared to other state-of-the-art methods.

cs.LG↗

arXiv2013

A Unified Framework for High-Dimensional Analysis of M-Estimators with Decomposable Regularizers

High-dimensional statistical inference deals with models in which the the number of parameters p is comparable to or larger than the sample size n. Since it is usually impossible to obtain consistent procedures unless $p/n\rightarrow0$, a line of recent work has studied models with various types of low-dimensional structure, including sparse vectors, sparse and structured matrices, low-rank matrices and combinations thereof. In such settings, a general approach to estimation is to solve a regularized optimization problem, which combines a loss function measuring how well the model fits the data with some regularization function that encourages the assumed structure. This paper provides a unified framework for establishing consistency and convergence rates for such regularized M-estimators under high-dimensional scaling. We state one main theorem and show how it can be used to re-derive some existing results, and also to obtain a number of new results on consistency and convergence rates, in both $\ell_2$-error and related norms. Our analysis also identifies two key properties of loss and regularization functions, referred to as restricted strong convexity and decomposability, that ensure corresponding regularized M-estimators have fast convergence rates and which are optimal in many well-studied cases.

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arXiv2012

A Hierarchical Graphical Model for Record Linkage

The task of matching co-referent records is known among other names as rocord linkage. For large record-linkage problems, often there is little or no labeled data available, but unlabeled data shows a reasonable clear structure. For such problems, unsupervised or semi-supervised methods are preferable to supervised methods. In this paper, we describe a hierarchical graphical model framework for the linakge-problem in an unsupervised setting. In addition to proposing new methods, we also cast existing unsupervised probabilistic record-linkage methods in this framework. Some of the techniques we propose to minimize overfitting in the above model are of interest in the general graphical model setting. We describe a method for incorporating monotinicity constraints in a graphical model. We also outline a bootstrapping approach of using "single-field" classifiers to noisily label latent variables in a hierarchical model. Experimental results show that our proposed unsupervised methods perform quite competitively even with fully supervised record-linkage methods.

cs.LG↗

arXiv2012

Variational Chernoff Bounds for Graphical Models

Recent research has made significant progress on the problem of bounding log partition functions for exponential family graphical models. Such bounds have associated dual parameters that are often used as heuristic estimates of the marginal probabilities required in inference and learning. However these variational estimates do not give rigorous bounds on marginal probabilities, nor do they give estimates for probabilities of more general events than simple marginals. In this paper we build on this recent work by deriving rigorous upper and lower bounds on event probabilities for graphical models. Our approach is based on the use of generalized Chernoff bounds to express bounds on event probabilities in terms of convex optimization problems; these optimization problems, in turn, require estimates of generalized log partition functions. Simulations indicate that this technique can result in useful, rigorous bounds to complement the heuristic variational estimates, with comparable computational cost.

cs.LG↗

arXiv2011

High-dimensional Sparse Inverse Covariance Estimation using Greedy Methods

In this paper we consider the task of estimating the non-zero pattern of the sparse inverse covariance matrix of a zero-mean Gaussian random vector from a set of iid samples. Note that this is also equivalent to recovering the underlying graph structure of a sparse Gaussian Markov Random Field (GMRF). We present two novel greedy approaches to solving this problem. The first estimates the non-zero covariates of the overall inverse covariance matrix using a series of global forward and backward greedy steps. The second estimates the neighborhood of each node in the graph separately, again using greedy forward and backward steps, and combines the intermediate neighborhoods to form an overall estimate. The principal contribution of this paper is a rigorous analysis of the sparsistency, or consistency in recovering the sparsity pattern of the inverse covariance matrix. Surprisingly, we show that both the local and global greedy methods learn the full structure of the model with high probability given just $O(d\log(p))$ samples, which is a \emph{significant} improvement over state of the art $\ell_1$-regularized Gaussian MLE (Graphical Lasso) that requires $O(d^2\log(p))$ samples. Moreover, the restricted eigenvalue and smoothness conditions imposed by our greedy methods are much weaker than the strong irrepresentable conditions required by the $\ell_1$-regularization based methods. We corroborate our results with extensive simulations and examples, comparing our local and global greedy methods to the $\ell_1$-regularized Gaussian MLE as well as the Neighborhood Greedy method to that of nodewise $\ell_1$-regularized linear regression (Neighborhood Lasso).

cs.LG↗

arXiv2011

Information-theoretic lower bounds on the oracle complexity of stochastic convex optimization

Relative to the large literature on upper bounds on complexity of convex optimization, lesser attention has been paid to the fundamental hardness of these problems. Given the extensive use of convex optimization in machine learning and statistics, gaining an understanding of these complexity-theoretic issues is important. In this paper, we study the complexity of stochastic convex optimization in an oracle model of computation. We improve upon known results and obtain tight minimax complexity estimates for various function classes.

stat.ML↗

arXiv2011

Virus Dynamics on Starlike Graphs

The field of epidemiology has presented fascinating and relevant questions for mathematicians, primarily concerning the spread of viruses in a community. The importance of this research has greatly increased over time as its applications have expanded to also include studies of electronic and social networks and the spread of information and ideas. We study virus propagation on a non-linear hub and spoke graph (which models well many airline networks). We determine the long-term behavior as a function of the cure and infection rates, as well as the number of spokes n. For each n we prove the existence of a critical threshold relating the two rates. Below this threshold, the virus always dies out; above this threshold, all non-trivial initial conditions iterate to a unique non-trivial steady state. We end with some generalizations to other networks.

math.DS↗

arXiv2011

Encoding and decoding V1 fMRI responses to natural images with sparse nonparametric models

Functional MRI (fMRI) has become the most common method for investigating the human brain. However, fMRI data present some complications for statistical analysis and modeling. One recently developed approach to these data focuses on estimation of computational encoding models that describe how stimuli are transformed into brain activity measured in individual voxels. Here we aim at building encoding models for fMRI signals recorded in the primary visual cortex of the human brain. We use residual analyses to reveal systematic nonlinearity across voxels not taken into account by previous models. We then show how a sparse nonparametric method [J. Roy. Statist. Soc. Ser. B 71 (2009b) 1009-1030] can be used together with correlation screening to estimate nonlinear encoding models effectively. Our approach produces encoding models that predict about 25% more accurately than models estimated using other methods [Nature 452 (2008a) 352--355]. The estimated nonlinearity impacts the inferred properties of individual voxels, and it has a plausible biological interpretation. One benefit of quantitative encoding models is that estimated models can be used to decode brain activity, in order to identify which specific image was seen by an observer. Encoding models estimated by our approach also improve such image identification by about 12% when the correct image is one of 11,500 possible images.

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