Optimal designs for Lasso and Dantzig selector using Expander Codes

TL;DR

This paper demonstrates that adjacency matrices of unbalanced expander graphs enable optimal high-dimensional regression using Lasso and Dantzig selector, with explicit constants and deterministic polynomial-time construction.

Contribution

It proves optimal error bounds for Lasso and Dantzig selector with expander graph matrices and provides a deterministic construction method.

Findings

Achieves near-oracle error bounds for high-dimensional regression.

Provides explicit restricted eigenvalue and compatibility constants.

Offers a polynomial-time deterministic construction of design matrices.

Abstract

We investigate the high-dimensional regression problem using adjacency matrices of unbalanced expander graphs. In this frame, we prove that the $\ell_{2}$-prediction error and the $\ell_{1}$-risk of the lasso and the Dantzig selector are optimal up to an explicit multiplicative constant. Thus we can estimate a high-dimensional target vector with an error term similar to the one obtained in a situation where one knows the support of the largest coordinates in advance. Moreover, we show that these design matrices have an explicit restricted eigenvalue. Precisely, they satisfy the restricted eigenvalue assumption and the compatibility condition with an explicit constant. Eventually, we capitalize on the recent construction of unbalanced expander graphs due to Guruswami, Umans, and Vadhan, to provide a deterministic polynomial time construction of these design matrices.