# An Efficient Linearly Convergent Regularized Proximal Point Algorithm for Fused Multiple Graphical Lasso Problems

## TL;DR

This paper introduces a new regularized proximal point algorithm with a superlinearly convergent semismooth Newton method for efficiently solving fused multiple graphical Lasso problems, improving convergence speed and robustness.

## Contribution

It develops an efficient second-order method leveraging explicit Jacobian expressions, outperforming existing first-order approaches in fused graphical model estimation.

## Key findings

- The proposed algorithm converges faster than state-of-the-art methods.
- It demonstrates robustness and efficiency on synthetic and real datasets.
- Explicit Jacobian derivation enhances the semismooth Newton method's performance.

## Abstract

Nowadays, analysing data from different classes or over a temporal grid has attracted a great deal of interest. As a result, various multiple graphical models for learning a collection of graphical models simultaneously have been derived by introducing sparsity in graphs and similarity across multiple graphs. This paper focuses on the fused multiple graphical Lasso model which encourages not only shared pattern of sparsity, but also shared values of edges across different graphs. For solving this model, we develop an efficient regularized proximal point algorithm, where the subproblem in each iteration of the algorithm is solved by a superlinearly convergent semismooth Newton method. To implement the semismooth Newton method, we derive an explicit expression for the generalized Jacobian of the proximal mapping of the fused multiple graphical Lasso regularizer. Unlike those widely used first order methods, our approach has heavily exploited the underlying second order information through the semismooth Newton method. This can not only accelerate the convergence of the algorithm, but also improve its robustness. The efficiency and robustness of our proposed algorithm are demonstrated by comparing with some state-of-the-art methods on both synthetic and real data sets. Supplementary materials for this article are available online.