Break recovery in graphical networks with D-trace loss

TL;DR

This paper introduces a new method for estimating time-varying sparse precision matrices in graphical networks using D-trace loss, addressing change-point detection and solution existence issues with a modified regularizer and ADMM algorithm.

Contribution

It proposes an alternative to Gaussian likelihood loss for change-point detection in dynamic networks, with theoretical consistency results and a practical solution approach.

Findings

The method accurately detects change-points in simulations.

The approach performs well on real data experiments.

The modified regularizer ensures solution existence under broader conditions.

Abstract

We consider the problem of estimating a time-varying sparse precision matrix, which is assumed to evolve in a piece-wise constant manner. Building upon the Group Fused LASSO and LASSO penalty functions, we estimate both the network structure and the change-points. We propose an alternative estimator to the commonly employed Gaussian likelihood loss, namely the D-trace loss. We provide the conditions for the consistency of the estimated change-points and of the sparse estimators in each block. We show that the solutions to the corresponding estimation problem exist when some conditions relating to the tuning parameters of the penalty functions are satisfied. Unfortunately, these conditions are not verifiable in general, posing challenges for tuning the parameters in practice. To address this issue, we introduce a modified regularizer and develop a revised problem that always admits solutions: these solutions can be used for detecting possible unsolvability of the original problem or obtaining a solution of the original problem otherwise. An alternating direction method of multipliers (ADMM) is then proposed to solve the revised problem. The relevance of the method is illustrated through simulations and real data experiments.