581 research outputs found
Sparse Inverse Covariance Estimation for Chordal Structures
In this paper, we consider the Graphical Lasso (GL), a popular optimization
problem for learning the sparse representations of high-dimensional datasets,
which is well-known to be computationally expensive for large-scale problems.
Recently, we have shown that the sparsity pattern of the optimal solution of GL
is equivalent to the one obtained from simply thresholding the sample
covariance matrix, for sparse graphs under different conditions. We have also
derived a closed-form solution that is optimal when the thresholded sample
covariance matrix has an acyclic structure. As a major generalization of the
previous result, in this paper we derive a closed-form solution for the GL for
graphs with chordal structures. We show that the GL and thresholding
equivalence conditions can significantly be simplified and are expected to hold
for high-dimensional problems if the thresholded sample covariance matrix has a
chordal structure. We then show that the GL and thresholding equivalence is
enough to reduce the GL to a maximum determinant matrix completion problem and
drive a recursive closed-form solution for the GL when the thresholded sample
covariance matrix has a chordal structure. For large-scale problems with up to
450 million variables, the proposed method can solve the GL problem in less
than 2 minutes, while the state-of-the-art methods converge in more than 2
hours
Graphical continuous Lyapunov models
The linear Lyapunov equation of a covariance matrix parametrizes the
equilibrium covariance matrix of a stochastic process. This parametrization can
be interpreted as a new graphical model class, and we show how the model class
behaves under marginalization and introduce a method for structure learning via
-penalized loss minimization. Our proposed method is demonstrated to
outperform alternative structure learning algorithms in a simulation study, and
we illustrate its application for protein phosphorylation network
reconstruction.Comment: 10 pages, 5 figure
iPACOSE: an iterative algorithm for the estimation of gene regulation networks
In the context of Gaussian Graphical Models (GGMs) with high-
dimensional small sample data, we present a simple procedure to esti-
mate partial correlations under the constraint that some of them are
strictly zero. This method can also be extended to covariance selection.
If the goal is to estimate a GGM, our new procedure can be applied
to re-estimate the partial correlations after a first graph has been esti-
mated in the hope to improve the estimation of non-zero coefficients. In
a simulation study, we compare our new covariance selection procedure
to existing methods and show that the re-estimated partial correlation
coefficients may be closer to the real values in important cases
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