Latent Variable Graphical Model Selection via Convex Optimization – Supplementary
نویسندگان
چکیده
1. Matrix perturbation bounds. Given a low-rank matrix we consider what happens to the invariant subspaces when the matrix is perturbed by a small amount. We assume without loss of generality that the matrix under consideration is square and symmetric, and our methods can be extended to the general non-symmetric non-square case. We refer the interested reader to [1, 3] for more details, as the results presented here are only a brief summary of what is relevant for this paper. In particular the arguments presented here are along the lines of those presented in [1]. The appendices in [1] also provide a more refined analysis of second-order perturbation errors. The resolvent of a matrix M is given by (M − ζI)−1 [3], and it is welldefined for all ζ ∈ C that do not coincide with an eigenvalue of M . If M has no eigenvalue with magnitude equal to η, then we have by the Cauchy residue formula that the projector onto the invariant subspace of a matrix M corresponding to all singular values smaller than η is given by
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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...
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تاریخ انتشار 2011