Estimating Multiple Precision Matrices with Cluster Fusion Regularization
This work addresses the challenge of precision matrix estimation in multi-class settings where relationships are unknown, offering a practical solution for statistical modeling, though it is incremental as it builds on existing penalized likelihood methods.
The authors tackled the problem of estimating multiple precision matrices from different classes without prior knowledge of their relationships, proposing a penalized likelihood framework that simultaneously estimates the matrices and their relationships, showing in simulations and real data that it outperforms methods ignoring relationships and matches those using prior information.
We propose a penalized likelihood framework for estimating multiple precision matrices from different classes. Most existing methods either incorporate no information on relationships between the precision matrices, or require this information be known a priori. The framework proposed in this article allows for simultaneous estimation of the precision matrices and relationships between the precision matrices, jointly. Sparse and non-sparse estimators are proposed, both of which require solving a non-convex optimization problem. To compute our proposed estimators, we use an iterative algorithm which alternates between a convex optimization problem solved by blockwise coordinate descent and a k-means clustering problem. Blockwise updates for computing the sparse estimator require solving an elastic net penalized precision matrix estimation problem, which we solve using a proximal gradient descent algorithm. We prove that this subalgorithm has a linear rate of convergence. In simulation studies and two real data applications, we show that our method can outperform competitors that ignore relevant relationships between precision matrices and performs similarly to methods which use prior information often uknown in practice.