List-Decodable Sparse Mean Estimation via Difference-of-Pairs Filtering

TL;DR

This paper introduces a new, simpler method for list-decodable sparse mean estimation, providing the first efficient algorithms with strong guarantees for distributions with bounded moments, light tails, and Gaussian inliers.

Contribution

It develops a novel technique for list-decodable mean estimation and offers the first efficient algorithms with provable guarantees for sparse means under broad conditions.

Findings

Achieves error of (1/α)^{O(1/t)} for distributions with bounded moments.

Provides optimal error of Θ(√log(1/α)) for Gaussian inliers.

Establishes nearly-matching lower bounds for statistical query and polynomial testing.

Abstract

We study the problem of list-decodable sparse mean estimation. Specifically, for a parameter $\alpha \in (0, 1/2)$, we are given $m$ points in $\mathbb{R}^n$, $\lfloor \alpha m \rfloor$ of which are i.i.d. samples from a distribution $D$ with unknown $k$-sparse mean $\mu$. No assumptions are made on the remaining points, which form the majority of the dataset. The goal is to return a small list of candidates containing a vector $\widehat \mu$ such that $\| \widehat \mu - \mu \|_2$ is small. Prior work had studied the problem of list-decodable mean estimation in the dense setting. In this work, we develop a novel, conceptually simpler technique for list-decodable mean estimation. As the main application of our approach, we provide the first sample and computationally efficient algorithm for list-decodable sparse mean estimation. In particular, for distributions with "certifiably bounded" $t$-th moments in $k$-sparse directions and sufficiently light tails, our algorithm achieves error of $(1/\alpha)^{O(1/t)}$ with sample complexity $m = (k\log(n))^{O(t)}/\alpha$ and running time $\mathrm{poly}(mn^t)$. For the special case of Gaussian inliers, our algorithm achieves the optimal error guarantee of $\Theta (\sqrt{\log(1/\alpha)})$ with quasi-polynomial sample and computational complexity. We complement our upper bounds with nearly-matching statistical query and low-degree polynomial testing lower bounds.