DNNLasso: Scalable Graph Learning for Matrix-Variate Data
Meixia Lin, Yangjing Zhang
TL;DR
This work addresses scalable learning of row- and column-wise dependencies in matrix-variate observations by modeling the precision matrix as a KS sum, $\Sigma^{-1}=\Omega\oplus\Gamma$, which reduces dimensionality and preserves structure. The authors propose DNNLasso, a diagonally non-negative graphical lasso that uses an ADMM framework with explicit proximal operators for the KS log-determinant, yielding robust and fast optimization. Key contributions include the diagonally non-negative constraint to resolve identifiability and ensure bounded solutions, a provably convergent ADMM algorithm, and closed-form proximal updates that enable scalable learning on large graphs. Empirical results on synthetic data and COIL100 video data demonstrate that DNNLasso outperforms state-of-the-art KS estimators in both accuracy and computational efficiency, offering a practical tool for structure learning in high-dimensional matrix-variate settings.
Abstract
We consider the problem of jointly learning row-wise and column-wise dependencies of matrix-variate observations, which are modelled separately by two precision matrices. Due to the complicated structure of Kronecker-product precision matrices in the commonly used matrix-variate Gaussian graphical models, a sparser Kronecker-sum structure was proposed recently based on the Cartesian product of graphs. However, existing methods for estimating Kronecker-sum structured precision matrices do not scale well to large scale datasets. In this paper, we introduce DNNLasso, a diagonally non-negative graphical lasso model for estimating the Kronecker-sum structured precision matrix, which outperforms the state-of-the-art methods by a large margin in both accuracy and computational time. Our code is available at https://github.com/YangjingZhang/DNNLasso.