List-Decodable Sparse Mean Estimation via Difference-of-Pairs Filtering
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arXiv
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| Main Authors: | , , , , |
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| Format: | Preprint |
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2022
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| _version_ | 1866910513120673792 |
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| author | Diakonikolas, Ilias Kane, Daniel M. Karmalkar, Sushrut Pensia, Ankit Pittas, Thanasis |
| author_facet | Diakonikolas, Ilias Kane, Daniel M. Karmalkar, Sushrut Pensia, Ankit Pittas, Thanasis |
| contents | We study the problem of list-decodable sparse mean estimation. Specifically, for a parameter $α\in (0, 1/2)$, we are given $m$ points in $\mathbb{R}^n$, $\lfloor αm \rfloor$ of which are i.i.d. samples from a distribution $D$ with unknown $k$-sparse mean $μ$. 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 $\widehat μ$ such that $\| \widehat μ- μ\|_2$ 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 $(1/α)^{O(1/t)}$ with sample complexity $m = (k\log(n))^{O(t)}/α$ and running time $\mathrm{poly}(mn^t)$. For the special case of Gaussian inliers, our algorithm achieves the optimal error guarantee of $Θ(\sqrt{\log(1/α)})$ with quasi-polynomial sample and computational complexity. We complement our upper bounds with nearly-matching statistical query and low-degree polynomial testing lower bounds. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2206_05245 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | List-Decodable Sparse Mean Estimation via Difference-of-Pairs Filtering Diakonikolas, Ilias Kane, Daniel M. Karmalkar, Sushrut Pensia, Ankit Pittas, Thanasis Data Structures and Algorithms Machine Learning Statistics Theory We study the problem of list-decodable sparse mean estimation. Specifically, for a parameter $α\in (0, 1/2)$, we are given $m$ points in $\mathbb{R}^n$, $\lfloor αm \rfloor$ of which are i.i.d. samples from a distribution $D$ with unknown $k$-sparse mean $μ$. 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 $\widehat μ$ such that $\| \widehat μ- μ\|_2$ 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 $(1/α)^{O(1/t)}$ with sample complexity $m = (k\log(n))^{O(t)}/α$ and running time $\mathrm{poly}(mn^t)$. For the special case of Gaussian inliers, our algorithm achieves the optimal error guarantee of $Θ(\sqrt{\log(1/α)})$ with quasi-polynomial sample and computational complexity. We complement our upper bounds with nearly-matching statistical query and low-degree polynomial testing lower bounds. |
| title | List-Decodable Sparse Mean Estimation via Difference-of-Pairs Filtering |
| topic | Data Structures and Algorithms Machine Learning Statistics Theory |
| url | https://arxiv.org/abs/2206.05245 |