Provable Guarantees for Sparsity Recovery with Deterministic Missing Data Patterns

TL;DR

This paper presents a method for recovering the sparsity pattern of regression parameters from deterministic missing data using Lasso, with theoretical guarantees under certain conditions, addressing challenges beyond random missingness.

Contribution

It introduces an efficient imputation algorithm leveraging topological properties of the missing data pattern and provides theoretical guarantees for exact sparsity recovery.

Findings

Exact recovery achieved with high probability under specific conditions

Polynomial time and logarithmic sample complexity for the proposed method

Effective handling of deterministic, non-uniform missing data patterns

Abstract

We study the problem of consistently recovering the sparsity pattern of a regression parameter vector from correlated observations governed by deterministic missing data patterns using Lasso. We consider the case in which the observed dataset is censored by a deterministic, non-uniform filter. Recovering the sparsity pattern in datasets with deterministic missing structure can be arguably more challenging than recovering in a uniformly-at-random scenario. In this paper, we propose an efficient algorithm for missing value imputation by utilizing the topological property of the censorship filter. We then provide novel theoretical results for exact recovery of the sparsity pattern using the proposed imputation strategy. Our analysis shows that, under certain statistical and topological conditions, the hidden sparsity pattern can be recovered consistently with high probability in polynomial time and logarithmic sample complexity.