About: Does the $/ell_1$-norm Learn a Sparse Graph under Laplacian Constrained Graphical Models?

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An Entity of Type : schema:ScholarlyArticle, within Data Space : covidontheweb.inria.fr associated with source document(s)

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type	Academic Article research paper schema:ScholarlyArticle
isDefinedBy	Covid-on-the-Web dataset
title	Does the $/ell_1$-norm Learn a Sparse Graph under Laplacian Constrained Graphical Models?
Creator	Cardoso, M Palomar, Daniel Vinícius, José Ying, Jiaxi
source	ArXiv
abstract	We consider the problem of learning a sparse graph under Laplacian constrained Gaussian graphical models. This problem can be formulated as a penalized maximum likelihood estimation of the precision matrix under Laplacian structural constraints. Like in the classical graphical lasso problem, recent works made use of the $/ell_1$-norm regularization with the goal of promoting sparsity in Laplacian structural precision matrix estimation. However, we find that the widely used $/ell_1$-norm is not effective in imposing a sparse solution in this problem. Through empirical evidence, we observe that the number of nonzero graph weights grows with the increase of the regularization parameter. From a theoretical perspective, we prove that a large regularization parameter will surprisingly lead to a fully connected graph. To address this issue, we propose a nonconvex estimation method by solving a sequence of weighted $/ell_1$-norm penalized sub-problems and prove that the statistical error of the proposed estimator matches the minimax lower bound. To solve each sub-problem, we develop a projected gradient descent algorithm that enjoys a linear convergence rate. Numerical experiments involving synthetic and real-world data sets from the recent COVID-19 pandemic and financial stock markets demonstrate the effectiveness of the proposed method. An open source $/mathsf{R}$ package containing the code for all the experiments is available at https://github.com/mirca/sparseGraph.
has issue date	2020-06-26(xsd:dateTime)
has license	arxiv
sha1sum (hex)	9642b2b5a46b8a627ce410e86fbb1f60d08156a0
resource representing a document's title	Does the $/ell_1$-norm Learn a Sparse Graph under Laplacian Constrained Graphical Models?
resource representing a document's body	covid:9642b2b5a46b8a627ce410e86fbb1f60d08156a0#body_text
is schema:about of	named entity 'regularization' named entity 'estimator' named entity 'theoretical perspective' named entity 'Laplacian' named entity 'linear convergence' »more»

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