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

• URL: http://arxiv.org/abs/2206.05245v2
• Date: Fri, 5 Jul 2024 17:57:31 GMT
• Title: List-Decodable Sparse Mean Estimation via Difference-of-Pairs Filtering
• Authors: Ilias Diakonikolas, Daniel M. Kane, Sushrut Karmalkar, Ankit Pensia, Thanasis Pittas,
• Abstract summary: We develop a novel, conceptually simpler technique for list-decodable sparse mean estimation. In particular, for distributions with "certifiably bounded" $t-th moments in $k$-sparse directions, our algorithm achieves error of $(1/alpha)O (1/t)$ with sample complexity $m = (klog(n))O(t)/alpha(mnt)$. For the special case of Gaussian inliers, our algorithm achieves the optimal error guarantee of $Theta (sqrtlog
• Score: 42.526664955704746
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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