A Sharp Algorithmic Analysis of Covariate Adjusted Precision Matrix Estimation with General Structural Priors

TL;DR

This paper provides a detailed analysis of a class of gradient descent algorithms for high-dimensional covariate adjusted precision matrix estimation, showing they achieve optimal rates without impractical assumptions.

Contribution

It offers a sharp convergence and statistical rate analysis for these algorithms, removing previous resampling assumptions and revealing a time-data tradeoff.

Findings

Algorithms converge linearly even without convexity

Achieve minimax optimal statistical rates

Remove impractical resampling assumptions

Abstract

In this paper, we present a sharp analysis for a class of alternating projected gradient descent algorithms which are used to solve the covariate adjusted precision matrix estimation problem in the high-dimensional setting. We demonstrate that these algorithms not only enjoy a linear rate of convergence in the absence of convexity, but also attain the optimal statistical rate (i.e., minimax rate). By introducing the generic chaining, our analysis removes the impractical resampling assumption used in the previous work. Moreover, our results also reveal a time-data tradeoff in this covariate adjusted precision matrix estimation problem. Numerical experiments are provided to verify our theoretical results.