Latent Variable Graphical Model Selection via Convex Optimization
نویسنده
چکیده
I want to start by congratulating Professors Chandrasekaran, Parrilo and Willsky for this fine piece of work. Their paper, hereafter referred to as CPW, addresses one of the biggest practical challenges of Gaussian graphical models—how to make inferences for a graphical model in the presence of missing variables. The difficulty comes from the fact that the validity of conditional independence relationships implied by a graphical model relies critically on the assumption that all conditional variables are observed, which of course can be unrealistic. As CPW shows, this is not as hopeless as it might appear to be. They characterize conditions under which a conditional graphical model can be identified, and offer a penalized likelihood method to reconstruct it. CPW notes that with missing variables, the concentration matrix of the observables can be expressed as the difference between a sparse matrix and a low-rank matrix; and suggests to exploit the sparsity using an l1 penalty and the low-rank structure by a trace norm penalty. In particular, the trace norm penalty or, more generally, nuclear norm penalties, can be viewed as a convex relaxation to the more direct rank constraint. Its use oftentimes comes as a necessity because rank constrained optimization could be computationally prohibitive. Interestingly, as I note here, the current problem actually lends itself to efficient algorithms in dealing with the rank constraint, and therefore allows for an attractive alternative to the approach of CPW.
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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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