Of ” Latent Variable Graphical Model Selection via Convex Optimization
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چکیده
Since recently, there have been an increasing interest in the problem of estimating a high-dimensional matrix K that can be decomposed in a sum of a sparse matrix S∗ (i.e., a matrix having only a small number of nonzero entries) and a low rank matrix L∗. This is motivated by applications in computer vision, video segmentation, computational biology, semantic indexing etc. The main contribution and novelty of Chandrasekaran, Parrilo and Willsky paper (CPW in what follows) is to propose and study a method of inference about such decomposable matrices for a particular setting where K is the precision (concentration) matrix of a partially observed sparse Gaussian graphical model (GGM). In this case, K is the inverse of the covariance matrix of a Gaussian vector XO extracted from a larger Gaussian vector (XO, XH) with sparse inverse covariance matrix. Then it is easy to see that K can be represented as a sum of a sparse precision matrix S∗ corresponding to the observed variables XO and a matrix L∗ with rank at most h, where h is the dimension of the latent variables XH . If h is small, which is a typical situation in practice, then L∗ has low rank. The GGM with latent variables is of major interest for applications in biology or in social networks where one often does not observe all the variables relevant for depicting sparsely the conditional dependencies. Note that formally this is just one possible motivation and mathematically the problem is dealt with in more generality, namely, postulating that the precision matrix satisfies
منابع مشابه
Rejoinder: Latent Variable Graphical Model Selection via Convex Optimization by Venkat Chandrasekaran,
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Discussion of “Latent Variable Graphical Model Selection via Convex Optimization”
We wish to congratulate the authors for their innovative contribution, which is bound to inspire much further research. We find latent variable model selection to be a fantastic application of matrix decomposition methods, namely, the superposition of low-rank and sparse elements. Clearly, the methodology introduced in this paper is of potential interest across many disciplines. In the followin...
متن کاملDiscussion: Latent Variable Graphical Model Selection via Convex Optimization by Steffen Lauritzen
We want to congratulate the authors for a thought-provoking and very interesting paper. Sparse modeling of the concentration matrix has enjoyed popularity in recent years. It has been framed as a computationally convenient convex 1constrained estimation problem in Yuan and Lin (2007) and can be applied readily to higher-dimensional problems. The authors argue—we think correctly—that the sparsit...
متن کاملDiscussion: Latent variable graphical model selection via convex optimization
We want to congratulate the authors for a thought-provoking and very interesting paper. Sparse modeling of the concentration matrix has enjoyed popularity in recent years. It has been framed as a computationally convenient convex l1-constrained estimation problem in Yuan and Lin (2007) and can be applied readily to higher-dimensional problems. The authors argue— we think correctly—that the spar...
متن کاملLatent Variable Graphical Model Selection via Convex Optimization1 by Venkat Chandrasekaran,
Suppose we observe samples of a subset of a collection of random variables. No additional information is provided about the number of latent variables, nor of the relationship between the latent and observed variables. Is it possible to discover the number of latent components, and to learn a statistical model over the entire collection of variables? We address this question in the setting in w...
متن کاملDiscussion : Latent Variable Graphical Model Selection via Convex Optimization
1. Introduction. We would like to congratulate the authors for their refreshing contribution to this high-dimensional latent variables graphical model selection problem. The problem of covariance and concentration matrices is fundamentally important in several classical statistical methodolo-gies and many applications. Recently, sparse concentration matrices estimation had received considerable...
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تاریخ انتشار 2012