Change Point Detection in Precision Matrices with D-trace Loss
研究如何估计随时间分段常数变化的稀疏精度矩阵，提出用D-trace损失替代高斯似然损失，并给出变点一致性和稀疏估计的条件，通过ADMM算法求解修正后的优化问题。
We consider the problem of estimating a time-varying sparse precision matrix, which is assumed to evolve in a piecewise constant manner. Building upon the Group Fused LASSO and LASSO penalty functions, we estimate both the precision matrix 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 numerical experiments.