Restricted Eigenvalue Conditions on Subgaussian Random Matrices

TL;DR

This paper investigates conditions under which certain subgaussian random matrices satisfy the Restricted Eigenvalue (RE) condition, linking geometric complexity with sample size requirements and broadening the class of matrices known to meet these conditions.

Contribution

It introduces an explicit covariance structure to broad classes of random matrices satisfying the RE condition, extending previous results related to the Restricted Isometry Property.

Findings

Random matrices with independent rows satisfy RE condition above a sample size threshold.

The introduced covariance structure broadens the class of matrices satisfying RE.

Geometric functional analysis is crucial for understanding sample complexity and high-dimensional implications.

Abstract

It is natural to ask: what kinds of matrices satisfy the Restricted Eigenvalue (RE) condition? In this paper, we associate the RE condition (Bickel-Ritov-Tsybakov 09) with the complexity of a subset of the sphere in $\R^p$, where $p$ is the dimensionality of the data, and show that a class of random matrices with independent rows, but not necessarily independent columns, satisfy the RE condition, when the sample size is above a certain lower bound. Here we explicitly introduce an additional covariance structure to the class of random matrices that we have known by now that satisfy the Restricted Isometry Property as defined in Candes and Tao 05 (and hence the RE condition), in order to compose a broader class of random matrices for which the RE condition holds. In this case, tools from geometric functional analysis in characterizing the intrinsic low-dimensional structures associated with the RE condition has been crucial in analyzing the sample complexity and understanding its statistical implications for high dimensional data.