This paper presents a convex-analytic framework to learn sparse graphs from data. While our problem formulation is inspired by an extension of the graphical lasso using the so-called combinatorial graph Laplacian framework, a key difference is the use of a nonconvex alternative to the l1 norm to attain graphs with better interpretability. Specifically, we use the weakly-convex minimax concave penalty (the difference between the l1 norm and the Huber function) which is known to yield sparse solutions with lower estimation bias than l1 for regression problems. In our framework, the graph Laplacian is replaced in the optimization by a linear transform of the vector corresponding to its upper triangular part. Via a reformulation relying on Moreau's decomposition, we show that overall convexity is guaranteed by introducing a quadratic function to our cost function. The problem can be solved efficiently by the primal-dual splitting method, of which the admissible conditions for provable convergence are presented. Numerical examples show that the proposed method significantly outperforms the existing graph learning methods with reasonable computation time.
Tatsuya KOYAKUMARU
Keio University
Masahiro YUKAWA
Keio University
Eduardo PAVEZ
University of Southern California
Antonio ORTEGA
University of Southern California
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Tatsuya KOYAKUMARU, Masahiro YUKAWA, Eduardo PAVEZ, Antonio ORTEGA, "Learning Sparse Graph with Minimax Concave Penalty under Gaussian Markov Random Fields" in IEICE TRANSACTIONS on Fundamentals,
vol. E106-A, no. 1, pp. 23-34, January 2023, doi: 10.1587/transfun.2021EAP1153.
Abstract: This paper presents a convex-analytic framework to learn sparse graphs from data. While our problem formulation is inspired by an extension of the graphical lasso using the so-called combinatorial graph Laplacian framework, a key difference is the use of a nonconvex alternative to the l1 norm to attain graphs with better interpretability. Specifically, we use the weakly-convex minimax concave penalty (the difference between the l1 norm and the Huber function) which is known to yield sparse solutions with lower estimation bias than l1 for regression problems. In our framework, the graph Laplacian is replaced in the optimization by a linear transform of the vector corresponding to its upper triangular part. Via a reformulation relying on Moreau's decomposition, we show that overall convexity is guaranteed by introducing a quadratic function to our cost function. The problem can be solved efficiently by the primal-dual splitting method, of which the admissible conditions for provable convergence are presented. Numerical examples show that the proposed method significantly outperforms the existing graph learning methods with reasonable computation time.
URL: https://global.ieice.org/en_transactions/fundamentals/10.1587/transfun.2021EAP1153/_p
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@ARTICLE{e106-a_1_23,
author={Tatsuya KOYAKUMARU, Masahiro YUKAWA, Eduardo PAVEZ, Antonio ORTEGA, },
journal={IEICE TRANSACTIONS on Fundamentals},
title={Learning Sparse Graph with Minimax Concave Penalty under Gaussian Markov Random Fields},
year={2023},
volume={E106-A},
number={1},
pages={23-34},
abstract={This paper presents a convex-analytic framework to learn sparse graphs from data. While our problem formulation is inspired by an extension of the graphical lasso using the so-called combinatorial graph Laplacian framework, a key difference is the use of a nonconvex alternative to the l1 norm to attain graphs with better interpretability. Specifically, we use the weakly-convex minimax concave penalty (the difference between the l1 norm and the Huber function) which is known to yield sparse solutions with lower estimation bias than l1 for regression problems. In our framework, the graph Laplacian is replaced in the optimization by a linear transform of the vector corresponding to its upper triangular part. Via a reformulation relying on Moreau's decomposition, we show that overall convexity is guaranteed by introducing a quadratic function to our cost function. The problem can be solved efficiently by the primal-dual splitting method, of which the admissible conditions for provable convergence are presented. Numerical examples show that the proposed method significantly outperforms the existing graph learning methods with reasonable computation time.},
keywords={},
doi={10.1587/transfun.2021EAP1153},
ISSN={1745-1337},
month={January},}
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TY - JOUR
TI - Learning Sparse Graph with Minimax Concave Penalty under Gaussian Markov Random Fields
T2 - IEICE TRANSACTIONS on Fundamentals
SP - 23
EP - 34
AU - Tatsuya KOYAKUMARU
AU - Masahiro YUKAWA
AU - Eduardo PAVEZ
AU - Antonio ORTEGA
PY - 2023
DO - 10.1587/transfun.2021EAP1153
JO - IEICE TRANSACTIONS on Fundamentals
SN - 1745-1337
VL - E106-A
IS - 1
JA - IEICE TRANSACTIONS on Fundamentals
Y1 - January 2023
AB - This paper presents a convex-analytic framework to learn sparse graphs from data. While our problem formulation is inspired by an extension of the graphical lasso using the so-called combinatorial graph Laplacian framework, a key difference is the use of a nonconvex alternative to the l1 norm to attain graphs with better interpretability. Specifically, we use the weakly-convex minimax concave penalty (the difference between the l1 norm and the Huber function) which is known to yield sparse solutions with lower estimation bias than l1 for regression problems. In our framework, the graph Laplacian is replaced in the optimization by a linear transform of the vector corresponding to its upper triangular part. Via a reformulation relying on Moreau's decomposition, we show that overall convexity is guaranteed by introducing a quadratic function to our cost function. The problem can be solved efficiently by the primal-dual splitting method, of which the admissible conditions for provable convergence are presented. Numerical examples show that the proposed method significantly outperforms the existing graph learning methods with reasonable computation time.
ER -