Minimax Estimation of Bandable Precision Matrices

TL;DR

This paper establishes improved minimax estimation bounds for banded precision matrices in multivariate normal models, showing that the estimation rate matches that of banded covariance matrices, with theoretical and experimental validation.

Contribution

It introduces the first minimax bounds for banded precision matrices under the spectral norm, matching the rates for covariance matrices, and provides a novel analysis technique involving subblock inversion.

Findings

Minimax rate for banded precision matrices matches that of covariance matrices.

Proposed method achieves nearly-noisy estimates of subblocks via covariance matrix inversion.

Experimental results confirm the theoretical bounds' sharpness.

Abstract

The inverse covariance matrix provides considerable insight for understanding statistical models in the multivariate setting. In particular, when the distribution over variables is assumed to be multivariate normal, the sparsity pattern in the inverse covariance matrix, commonly referred to as the precision matrix, corresponds to the adjacency matrix representation of the Gauss-Markov graph, which encodes conditional independence statements between variables. Minimax results under the spectral norm have previously been established for covariance matrices, both sparse and banded, and for sparse precision matrices. We establish minimax estimation bounds for estimating banded precision matrices under the spectral norm. Our results greatly improve upon the existing bounds; in particular, we find that the minimax rate for estimating banded precision matrices matches that of estimating banded covariance matrices. The key insight in our analysis is that we are able to obtain barely-noisy estimates of $k \times k$ subblocks of the precision matrix by inverting slightly wider blocks of the empirical covariance matrix along the diagonal. Our theoretical results are complemented by experiments demonstrating the sharpness of our bounds.