arXiv subjects

Arindam Banerjee

Publications and source records attributed to Arindam Banerjee.

98 records · Page 6Linked to original sources

arXiv2014

Bregman Alternating Direction Method of Multipliers

The mirror descent algorithm (MDA) generalizes gradient descent by using a Bregman divergence to replace squared Euclidean distance. In this paper, we similarly generalize the alternating direction method of multipliers (ADMM) to Bregman ADMM (BADMM), which allows the choice of different Bregman divergences to exploit the structure of problems. BADMM provides a unified framework for ADMM and its variants, including generalized ADMM, inexact ADMM and Bethe ADMM. We establish the global convergence and the $O(1/T)$ iteration complexity for BADMM. In some cases, BADMM can be faster than ADMM by a factor of $O(n/\log(n))$. In solving the linear program of mass transportation problem, BADMM leads to massive parallelism and can easily run on GPU. BADMM is several times faster than highly optimized commercial software Gurobi.

math.OC↗

arXiv2014

The Regularity of Powers of Edge Ideals

In this paper we prove the existence of a special order on the set of minimal monomial generators of powers of edge ideals of arbitrary graphs. Using this order we find new upper bounds on the regularity of powers of edge ideals of graphs whose complement does not have any induced four cycle.

math.AC↗

arXiv2013

Bethe-ADMM for Tree Decomposition based Parallel MAP Inference

We consider the problem of maximum a posteriori (MAP) inference in discrete graphical models. We present a parallel MAP inference algorithm called Bethe-ADMM based on two ideas: tree-decomposition of the graph and the alternating direction method of multipliers (ADMM). However, unlike the standard ADMM, we use an inexact ADMM augmented with a Bethe-divergence based proximal function, which makes each subproblem in ADMM easy to solve in parallel using the sum-product algorithm. We rigorously prove global convergence of Bethe-ADMM. The proposed algorithm is extensively evaluated on both synthetic and real datasets to illustrate its effectiveness. Further, the parallel Bethe-ADMM is shown to scale almost linearly with increasing number of cores.

cs.AI↗

arXiv2013

Online Alternating Direction Method (longer version)

Online optimization has emerged as powerful tool in large scale optimization. In this pa- per, we introduce efficient online optimization algorithms based on the alternating direction method (ADM), which can solve online convex optimization under linear constraints where the objective could be non-smooth. We introduce new proof techniques for ADM in the batch setting, which yields a O(1/T) convergence rate for ADM and forms the basis for regret anal- ysis in the online setting. We consider two scenarios in the online setting, based on whether an additional Bregman divergence is needed or not. In both settings, we establish regret bounds for both the objective function as well as constraints violation for general and strongly convex functions. We also consider inexact ADM updates where certain terms are linearized to yield efficient updates and show the stochastic convergence rates. In addition, we briefly discuss that online ADM can be used as projection- free online learning algorithm in some scenarios. Preliminary results are presented to illustrate the performance of the proposed algorithms.

cs.LG↗

arXiv2012

Gap Filling in the Plant Kingdom---Trait Prediction Using Hierarchical Probabilistic Matrix Factorization

Plant traits are a key to understanding and predicting the adaptation of ecosystems to environmental changes, which motivates the TRY project aiming at constructing a global database for plant traits and becoming a standard resource for the ecological community. Despite its unprecedented coverage, a large percentage of missing data substantially constrains joint trait analysis. Meanwhile, the trait data is characterized by the hierarchical phylogenetic structure of the plant kingdom. While factorization based matrix completion techniques have been widely used to address the missing data problem, traditional matrix factorization methods are unable to leverage the phylogenetic structure. We propose hierarchical probabilistic matrix factorization (HPMF), which effectively uses hierarchical phylogenetic information for trait prediction. We demonstrate HPMF's high accuracy, effectiveness of incorporating hierarchical structure and ability to capture trait correlation through experiments.

cs.CE↗

arXiv2012

Online Alternating Direction Method

Online optimization has emerged as powerful tool in large scale optimization. In this paper, we introduce efficient online algorithms based on the alternating directions method (ADM). We introduce a new proof technique for ADM in the batch setting, which yields the O(1/T) convergence rate of ADM and forms the basis of regret analysis in the online setting. We consider two scenarios in the online setting, based on whether the solution needs to lie in the feasible set or not. In both settings, we establish regret bounds for both the objective function as well as constraint violation for general and strongly convex functions. Preliminary results are presented to illustrate the performance of the proposed algorithms.

cs.LG↗

arXiv2012

Gaussian Process Topic Models

We introduce Gaussian Process Topic Models (GPTMs), a new family of topic models which can leverage a kernel among documents while extracting correlated topics. GPTMs can be considered a systematic generalization of the Correlated Topic Models (CTMs) using ideas from Gaussian Process (GP) based embedding. Since GPTMs work with both a topic covariance matrix and a document kernel matrix, learning GPTMs involves a novel component-solving a suitable Sylvester equation capturing both topic and document dependencies. The efficacy of GPTMs is demonstrated with experiments evaluating the quality of both topic modeling and embedding.

cs.LG↗

arXiv2009

Approximation Algorithms for Bregman Co-clustering and Tensor Clustering

In the past few years powerful generalizations to the Euclidean k-means problem have been made, such as Bregman clustering [7], co-clustering (i.e., simultaneous clustering of rows and columns of an input matrix) [9,18], and tensor clustering [8,34]. Like k-means, these more general problems also suffer from the NP-hardness of the associated optimization. Researchers have developed approximation algorithms of varying degrees of sophistication for k-means, k-medians, and more recently also for Bregman clustering [2]. However, there seem to be no approximation algorithms for Bregman co- and tensor clustering. In this paper we derive the first (to our knowledge) guaranteed methods for these increasingly important clustering settings. Going beyond Bregman divergences, we also prove an approximation factor for tensor clustering with arbitrary separable metrics. Through extensive experiments we evaluate the characteristics of our method, and show that it also has practical impact.

cs.DS↗