Discussion: Latent variable graphical model selection via convex optimization
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چکیده
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 following, we will first discuss this paper in more detail and then reflect on the versatility of the low-rank + sparse decomposition. Latent variable model selection. The proposed scheme is an extension of the graphical lasso of Yuan and Lin [15] (see also [1, 6]), which is a popular approach for learning the structure in an undirected Gaussian graphical model. In this setup, we assume we have independent samples X ∼ N (0, Σ) with a covariance matrix Σ exhibiting a sparse dependence structure but otherwise unknown; that is to say, most pairs of variables are conditionally independent given all the others. Formally, the concentration matrix Σ −1 is assumed to be sparse. A natural fitting procedure is then to regularize the likelihood by adding a term proportional to the ℓ 1 norm of the estimated inverse covariance matrix S: minimize −ℓ(S, Σ n 0) + λS 1 (1) under the constraint S 0, where Σ n 0 is the empirical covariance matrix and S 1 = ij |S ij |. (Variants are possible depending upon whether or not one would want to penalize the diagonal elements.) This problem is convex. When some variables are unobserved—the observed and hidden variables are still jointly Gaussian—the model above may not be appropriate because the hidden variables can have a confounding effect. An example is this: we observe stock prices of companies and would like to infer conditional
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 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