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Arindam Banerjee

Publications and source records attributed to Arindam Banerjee.

At least 91 records · Page 5Linked to original sources
arXiv2016

The Matrix Generalized Inverse Gaussian Distribution: Properties and Applications

While the Matrix Generalized Inverse Gaussian ($\mathcal{MGIG}$) distribution arises naturally in some settings as a distribution over symmetric positive semi-definite matrices, certain key properties of the distribution and effective ways of sampling from the distribution have not been carefully studied. In this paper, we show that the $\mathcal{MGIG}$ is unimodal, and the mode can be obtained by solving an Algebraic Riccati Equation (ARE) equation [7]. Based on the property, we propose an importance sampling method for the $\mathcal{MGIG}$ where the mode of the proposal distribution matches that of the target. The proposed sampling method is more efficient than existing approaches [32, 33], which use proposal distributions that may have the mode far from the $\mathcal{MGIG}$'s mode. Further, we illustrate that the the posterior distribution in latent factor models, such as probabilistic matrix factorization (PMF) [25], when marginalized over one latent factor has the $\mathcal{MGIG}$ distribution. The characterization leads to a novel Collapsed Monte Carlo (CMC) inference algorithm for such latent factor models. We illustrate that CMC has a lower log loss or perplexity than MCMC, and needs fewer samples.

stat.ML↗
arXiv2016

Alternating Estimation for Structured High-Dimensional Multi-Response Models

We consider learning high-dimensional multi-response linear models with structured parameters. By exploiting the noise correlations among responses, we propose an alternating estimation (AltEst) procedure to estimate the model parameters based on the generalized Dantzig selector. Under suitable sample size and resampling assumptions, we show that the error of the estimates generated by AltEst, with high probability, converges linearly to certain minimum achievable level, which can be tersely expressed by a few geometric measures, such as Gaussian width of sets related to the parameter structure. To the best of our knowledge, this is the first non-asymptotic statistical guarantee for such AltEst-type algorithm applied to estimation problem with general structures.

stat.ML↗
arXiv2016

Structured Stochastic Linear Bandits

The stochastic linear bandit problem proceeds in rounds where at each round the algorithm selects a vector from a decision set after which it receives a noisy linear loss parameterized by an unknown vector. The goal in such a problem is to minimize the (pseudo) regret which is the difference between the total expected loss of the algorithm and the total expected loss of the best fixed vector in hindsight. In this paper, we consider settings where the unknown parameter has structure, e.g., sparse, group sparse, low-rank, which can be captured by a norm, e.g., $L_1$, $L_{(1,2)}$, nuclear norm. We focus on constructing confidence ellipsoids which contain the unknown parameter across all rounds with high-probability. We show the radius of such ellipsoids depend on the Gaussian width of sets associated with the norm capturing the structure. Such characterization leads to tighter confidence ellipsoids and, therefore, sharper regret bounds compared to bounds in the existing literature which are based on the ambient dimensionality.

stat.ML↗
arXiv2016

Generalized Direct Change Estimation in Ising Model Structure

We consider the problem of estimating change in the dependency structure between two $p$-dimensional Ising models, based on respectively $n_1$ and $n_2$ samples drawn from the models. The change is assumed to be structured, e.g., sparse, block sparse, node-perturbed sparse, etc., such that it can be characterized by a suitable (atomic) norm. We present and analyze a norm-regularized estimator for directly estimating the change in structure, without having to estimate the structures of the individual Ising models. The estimator can work with any norm, and can be generalized to other graphical models under mild assumptions. We show that only one set of samples, say $n_2$, needs to satisfy the sample complexity requirement for the estimator to work, and the estimation error decreases as $\frac{c}{\sqrt{\min(n_1,n_2)}}$, where $c$ depends on the Gaussian width of the unit norm ball. For example, for $\ell_1$ norm applied to $s$-sparse change, the change can be accurately estimated with $\min(n_1,n_2)=O(s \log p)$ which is sharper than an existing result $n_1= O(s^2 \log p)$ and $n_2 = O(n_1^2)$. Experimental results illustrating the effectiveness of the proposed estimator are presented.

math.ST↗
arXiv2016

Graph Connectivity and Binomial Edge Ideals

We relate homological properties of a binomial edge ideal $\mathcal{J}_G$ to invariants that measure the connectivity of a simple graph $G$. Specifically, we show if $R/\mathcal{J}_G$ is a Cohen-Macaulay ring, then graph toughness of $G$ is exactly $\frac{1}{2}$. We also give an inequality between the depth of $R/\mathcal{J}_G$ and the vertex-connectivity of $G$. In addition, we study the Hilbert-Samuel multiplicity, and the Hilbert-Kunz multiplicity of $R/\mathcal{J}_G$.

math.AC↗
arXiv2016

Structured Matrix Recovery via the Generalized Dantzig Selector

In recent years, structured matrix recovery problems have gained considerable attention for its real world applications, such as recommender systems and computer vision. Much of the existing work has focused on matrices with low-rank structure, and limited progress has been made matrices with other types of structure. In this paper we present non-asymptotic analysis for estimation of generally structured matrices via the generalized Dantzig selector under generic sub-Gaussian measurements. We show that the estimation error can always be succinctly expressed in terms of a few geometric measures of suitable sets which only depend on the structure of the underlying true matrix. In addition, we derive the general bounds on these geometric measures for structures characterized by unitarily invariant norms, which is a large family covering most matrix norms of practical interest. Examples are provided to illustrate the utility of our theoretical development.

stat.ML↗
arXiv2016

A Spectral Algorithm for Inference in Hidden Semi-Markov Models

Hidden semi-Markov models (HSMMs) are latent variable models which allow latent state persistence and can be viewed as a generalization of the popular hidden Markov models (HMMs). In this paper, we introduce a novel spectral algorithm to perform inference in HSMMs. Unlike expectation maximization (EM), our approach correctly estimates the probability of given observation sequence based on a set of training sequences. Our approach is based on estimating moments from the sample, whose number of dimensions depends only logarithmically on the maximum length of the hidden state persistence. Moreover, the algorithm requires only a few matrix inversions and is therefore computationally efficient. Empirical evaluations on synthetic and real data demonstrate the advantage of the algorithm over EM in terms of speed and accuracy, especially for large datasets.

stat.ML↗
arXiv2016

Semi-Markov Switching Vector Autoregressive Model-based Anomaly Detection in Aviation Systems

In this work we consider the problem of anomaly detection in heterogeneous, multivariate, variable-length time series datasets. Our focus is on the aviation safety domain, where data objects are flights and time series are sensor readings and pilot switches. In this context the goal is to detect anomalous flight segments, due to mechanical, environmental, or human factors in order to identifying operationally significant events and provide insights into the flight operations and highlight otherwise unavailable potential safety risks and precursors to accidents. For this purpose, we propose a framework which represents each flight using a semi-Markov switching vector autoregressive (SMS-VAR) model. Detection of anomalies is then based on measuring dissimilarities between the model's prediction and data observation. The framework is scalable, due to the inherent parallel nature of most computations, and can be used to perform online anomaly detection. Extensive experimental results on simulated and real datasets illustrate that the framework can detect various types of anomalies along with the key parameters involved.

cs.LG↗
arXiv2016

Estimating Structured Vector Autoregressive Model

While considerable advances have been made in estimating high-dimensional structured models from independent data using Lasso-type models, limited progress has been made for settings when the samples are dependent. We consider estimating structured VAR (vector auto-regressive models), where the structure can be captured by any suitable norm, e.g., Lasso, group Lasso, order weighted Lasso, sparse group Lasso, etc. In VAR setting with correlated noise, although there is strong dependence over time and covariates, we establish bounds on the non-asymptotic estimation error of structured VAR parameters. Surprisingly, the estimation error is of the same order as that of the corresponding Lasso-type estimator with independent samples, and the analysis holds for any norm. Our analysis relies on results in generic chaining, sub-exponential martingales, and spectral representation of VAR models. Experimental results on synthetic data with a variety of structures as well as real aviation data are presented, validating theoretical results.

math.ST↗
arXiv2015

Estimation with Norm Regularization

Analysis of non-asymptotic estimation error and structured statistical recovery based on norm regularized regression, such as Lasso, needs to consider four aspects: the norm, the loss function, the design matrix, and the noise model. This paper presents generalizations of such estimation error analysis on all four aspects compared to the existing literature. We characterize the restricted error set where the estimation error vector lies, establish relations between error sets for the constrained and regularized problems, and present an estimation error bound applicable to any norm. Precise characterizations of the bound is presented for isotropic as well as anisotropic subGaussian design matrices, subGaussian noise models, and convex loss functions, including least squares and generalized linear models. Generic chaining and associated results play an important role in the analysis. A key result from the analysis is that the sample complexity of all such estimators depends on the Gaussian width of a spherical cap corresponding to the restricted error set. Further, once the number of samples $n$ crosses the required sample complexity, the estimation error decreases as $\frac{c}{\sqrt{n}}$, where $c$ depends on the Gaussian width of the unit norm ball.

stat.ML↗
arXiv2015

Enumerating all maximal biclusters in numerical datasets

Biclustering has proved to be a powerful data analysis technique due to its wide success in various application domains. However, the existing literature presents efficient solutions only for enumerating maximal biclusters with constant values, or heuristic-based approaches which can not find all biclusters or even support the maximality of the obtained biclusters. Here, we present a general family of biclustering algorithms for enumerating all maximal biclusters with (i) constant values on rows, (ii) constant values on columns, or (iii) coherent values. Versions for perfect and for perturbed biclusters are provided. Our algorithms have four key properties (just the algorithm for perturbed biclusters with coherent values fails to exhibit the first property): they are (1) efficient (take polynomial time per pattern), (2) complete (find all maximal biclusters), (3) correct (all biclusters attend the user-defined measure of similarity), and (4) non-redundant (all the obtained biclusters are maximal and the same bicluster is not enumerated twice). They are based on a generalization of an efficient formal concept analysis algorithm called In-Close2. Experimental results point to the necessity of having efficient enumerative biclustering algorithms and provide a valuable insight into the scalability of our family of algorithms and its sensitivity to user-defined parameters.

cs.DM↗
arXiv2015

Generalized Dantzig Selector: Application to the k-support norm

We propose a Generalized Dantzig Selector (GDS) for linear models, in which any norm encoding the parameter structure can be leveraged for estimation. We investigate both computational and statistical aspects of the GDS. Based on conjugate proximal operator, a flexible inexact ADMM framework is designed for solving GDS, and non-asymptotic high-probability bounds are established on the estimation error, which rely on Gaussian width of unit norm ball and suitable set encompassing estimation error. Further, we consider a non-trivial example of the GDS using $k$-support norm. We derive an efficient method to compute the proximal operator for $k$-support norm since existing methods are inapplicable in this setting. For statistical analysis, we provide upper bounds for the Gaussian widths needed in the GDS analysis, yielding the first statistical recovery guarantee for estimation with the $k$-support norm. The experimental results confirm our theoretical analysis.

stat.ML↗
arXiv2014

Properties of Lyubeznik numbers under localization and polarization

We exhibit a global bound for the Lyubeznik numbers of a ring of prime characteristic. In addition, we show that for a monomial ideal, the Lyubeznik numbers of the quotient rings of its radical and its polarization are the same. Furthermore, we present examples that show striking behavior of the Lyubeznik numbers under localization. We also show related results for generalizations of the Lyubeznik numbers.

math.AC↗
arXiv2014

Parallel Direction Method of Multipliers

We consider the problem of minimizing block-separable convex functions subject to linear constraints. While the Alternating Direction Method of Multipliers (ADMM) for two-block linear constraints has been intensively studied both theoretically and empirically, in spite of some preliminary work, effective generalizations of ADMM to multiple blocks is still unclear. In this paper, we propose a randomized block coordinate method named Parallel Direction Method of Multipliers (PDMM) to solve the optimization problems with multi-block linear constraints. PDMM randomly updates some primal and dual blocks in parallel, behaving like parallel randomized block coordinate descent. We establish the global convergence and the iteration complexity for PDMM with constant step size. We also show that PDMM can do randomized block coordinate descent on overlapping blocks. Experimental results show that PDMM performs better than state-of-the-arts methods in two applications, robust principal component analysis and overlapping group lasso.

math.OC↗
arXiv2014

Regularity of Path Ideals of Gap Free Graphs

In this paper we study the Castelnuovo-Mumford regularity of the path ideals of finite simple graphs. We find new upper bounds for various path ideals of gap free graphs. In particular we prove that the path ideals of gap free and claw graphs have linear minimal free resolutions.

math.AC↗
arXiv2014

Multi-task Sparse Structure Learning

Multi-task learning (MTL) aims to improve generalization performance by learning multiple related tasks simultaneously. While sometimes the underlying task relationship structure is known, often the structure needs to be estimated from data at hand. In this paper, we present a novel family of models for MTL, applicable to regression and classification problems, capable of learning the structure of task relationships. In particular, we consider a joint estimation problem of the task relationship structure and the individual task parameters, which is solved using alternating minimization. The task relationship structure learning component builds on recent advances in structure learning of Gaussian graphical models based on sparse estimators of the precision (inverse covariance) matrix. We illustrate the effectiveness of the proposed model on a variety of synthetic and benchmark datasets for regression and classification. We also consider the problem of combining climate model outputs for better projections of future climate, with focus on temperature in South America, and show that the proposed model outperforms several existing methods for the problem.

cs.LG↗
arXiv2014

Randomized Block Coordinate Descent for Online and Stochastic Optimization

Two types of low cost-per-iteration gradient descent methods have been extensively studied in parallel. One is online or stochastic gradient descent (OGD/SGD), and the other is randomzied coordinate descent (RBCD). In this paper, we combine the two types of methods together and propose online randomized block coordinate descent (ORBCD). At each iteration, ORBCD only computes the partial gradient of one block coordinate of one mini-batch samples. ORBCD is well suited for the composite minimization problem where one function is the average of the losses of a large number of samples and the other is a simple regularizer defined on high dimensional variables. We show that the iteration complexity of ORBCD has the same order as OGD or SGD. For strongly convex functions, by reducing the variance of stochastic gradients, we show that ORBCD can converge at a geometric rate in expectation, matching the convergence rate of SGD with variance reduction and RBCD.

cs.LG↗