Exact recovery in the double sparse model: sufficient and necessary signal conditions

TL;DR

This paper characterizes the precise signal conditions needed for exact support recovery in the double sparse model and proposes an optimal iterative hard thresholding method that achieves this under those conditions.

Contribution

It establishes the necessary and sufficient signal conditions for exact support recovery in the double sparse model and introduces an optimal algorithm that attains these conditions.

Findings

The proposed method achieves exact support recovery under the derived signal conditions.

No method can recover support if the signal conditions are not met.

Numerical experiments validate the theoretical results.

Abstract

The double sparse linear model, which has both group-wise and element-wise sparsity in regression coefficients, has attracted lots of attention recently. This paper establishes the sufficient and necessary relationship between the exact support recovery and the optimal minimum signal conditions in the double sparse model. Specifically, sharply under the proposed signal conditions, a two-stage double sparse iterative hard thresholding procedure achieves exact support recovery with a suitably chosen threshold parameter. Also, this procedure maintains asymptotic normality aligning with an OLS estimator given true support, hence holding the oracle properties. Conversely, we prove that no method can achieve exact support recovery if these signal conditions are violated. This fills a critical gap in the minimax optimality theory on support recovery of the double sparse model. Finally, numerical experiments are provided to support our theoretical findings.