Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Does the $\ell_1$-norm Learn a Sparse Graph under Laplacian Constrained
  Graphical Models?   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.

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The categories of this paper are: hep-th
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The categories of this paper are: q-fin
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The categories of this paper are: chem-ph
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The categories of this paper are: q-bio
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The categories of this paper are: funct-an
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The categories of this paper are: hep-ph
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The categories of this paper are: quant-ph
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The categories of this paper are: nucl-th
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The categories of this paper are: hep-lat
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The categories of this paper are: solv-int
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The categories of this paper are: acc-phys
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The categories of this paper are: cond-mat
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The categories of this paper are: supr-con
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The categories of this paper are: atom-ph
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The categories of this paper are: nlin
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The categories of this paper are: plasm-ph