A unified precision matrix estimation framework via sparse column-wise inverse operator under weak sparsity

TL;DR

This paper develops a unified framework for high-dimensional precision matrix estimation under weak sparsity, extending existing methods to handle diverse data types with comparable convergence rates.

Contribution

It introduces a general error bound for the SCIO estimator under weak sparsity and unifies approaches for heavy-tailed, non-paranormal, and matrix variate data.

Findings

Achieves optimal convergence rates similar to existing methods.

Handles various data types including heavy-tailed and non-paranormal.

Provides efficient implementation of the estimation procedures.

Abstract

In this paper, we estimate the high dimensional precision matrix under the weak sparsity condition where many entries are nearly zero. We revisit the sparse column-wise inverse operator (SCIO) estimator \cite{liu2015fast} and derive its general error bounds under the weak sparsity condition. A unified framework is established to deal with various cases including the heavy-tailed data, the non-paranormal data, and the matrix variate data. These new methods can achieve the same convergence rates as the existing methods and can be implemented efficiently.