High-dimensional Sparse Inverse Covariance Estimation using Greedy Methods

TL;DR

This paper introduces two greedy algorithms for high-dimensional sparse inverse covariance estimation, demonstrating they recover the true structure with fewer samples and under weaker conditions than traditional methods like Graphical Lasso.

Contribution

The paper proposes novel greedy approaches for inverse covariance estimation that require fewer samples and weaker assumptions, with rigorous theoretical guarantees and empirical validation.

Findings

Both greedy methods recover the full structure with O(d log p) samples.

Greedy methods outperform $ ext{l}_1$-regularized methods in sample efficiency.

Theoretical analysis shows weaker conditions are sufficient for consistency.

Abstract

In this paper we consider the task of estimating the non-zero pattern of the sparse inverse covariance matrix of a zero-mean Gaussian random vector from a set of iid samples. Note that this is also equivalent to recovering the underlying graph structure of a sparse Gaussian Markov Random Field (GMRF). We present two novel greedy approaches to solving this problem. The first estimates the non-zero covariates of the overall inverse covariance matrix using a series of global forward and backward greedy steps. The second estimates the neighborhood of each node in the graph separately, again using greedy forward and backward steps, and combines the intermediate neighborhoods to form an overall estimate. The principal contribution of this paper is a rigorous analysis of the sparsistency, or consistency in recovering the sparsity pattern of the inverse covariance matrix. Surprisingly, we show that both the local and global greedy methods learn the full structure of the model with high probability given just $O(d\log(p))$ samples, which is a \emph{significant} improvement over state of the art $\ell_1$-regularized Gaussian MLE (Graphical Lasso) that requires $O(d^2\log(p))$ samples. Moreover, the restricted eigenvalue and smoothness conditions imposed by our greedy methods are much weaker than the strong irrepresentable conditions required by the $\ell_1$-regularization based methods. We corroborate our results with extensive simulations and examples, comparing our local and global greedy methods to the $\ell_1$-regularized Gaussian MLE as well as the Neighborhood Greedy method to that of nodewise $\ell_1$-regularized linear regression (Neighborhood Lasso).