Minimax Rates for High-dimensional Double Sparse Structure over $\ell_u(\ell_q)$-balls

TL;DR

This paper establishes minimax rates for high-dimensional double sparse structures over $\,ell_u(\,ell_q)$-balls, revealing phase transitions, and introduces an optimal algorithm for double sparse linear regression.

Contribution

It develops a novel lower bound technique for the metric entropy of double sparse structures, proves matching upper bounds, and introduces the DSIHT algorithm for optimal estimation.

Findings

Discovered phase transition in minimax rates for $u,q \,\in\, (0,1]$.

Established minimax rates for double sparse regression.

Proposed the DSIHT algorithm with optimality in minimax sense.

Abstract

In this paper, we focus on the high-dimensional double sparse structure, where the parameter of interest simultaneously encourages group-wise sparsity and element-wise sparsity in each group. By combining the Gilbert-Varshamov bound and its variants, we develop a novel lower bound technique for the metric entropy of the parameter space, specifically tailored for the double sparse structure over $\ell_u(\ell_q)$-balls with $u,q \in [0,1]$. We prove lower bounds on the estimation error using an information-theoretic approach, leveraging our proposed lower bound technique and Fano's inequality. To complement the lower bounds, we establish matching upper bounds through a direct analysis of constrained least-squares estimators and utilize results from empirical processes. A significant finding of our study is the discovery of a phase transition phenomenon in the minimax rates for $u,q \in (0, 1]$. Furthermore, we extend the theoretical results to the double sparse regression model and determine its minimax rate for estimation error. To tackle double sparse linear regression, we develop the DSIHT (Double Sparse Iterative Hard Thresholding) algorithm, demonstrating its optimality in the minimax sense. Finally, we demonstrate the superiority of our method through numerical experiments.