Sparse permutation invariant covariance estimation

TL;DR

This paper introduces a permutation-invariant, sparse inverse covariance estimator for high-dimensional data, utilizing a penalized likelihood with a fast algorithm, and demonstrates its effectiveness through theoretical analysis and empirical tests.

Contribution

It presents a novel sparse inverse covariance estimator that is permutation-invariant, with proven convergence rates and an efficient algorithm, improving upon existing methods.

Findings

The estimator achieves favorable convergence rates depending on sparsity.

Correlation-based version improves operator norm rates.

Empirical results show competitive performance on simulated and real data.

Abstract

The paper proposes a method for constructing a sparse estimator for the inverse covariance (concentration) matrix in high-dimensional settings. The estimator uses a penalized normal likelihood approach and forces sparsity by using a lasso-type penalty. We establish a rate of convergence in the Frobenius norm as both data dimension $p$ and sample size $n$ are allowed to grow, and show that the rate depends explicitly on how sparse the true concentration matrix is. We also show that a correlation-based version of the method exhibits better rates in the operator norm. We also derive a fast iterative algorithm for computing the estimator, which relies on the popular Cholesky decomposition of the inverse but produces a permutation-invariant estimator. The method is compared to other estimators on simulated data and on a real data example of tumor tissue classification using gene expression data.