List-decodable robust mean estimation and learning mixtures of spherical gaussians
List-decodable robust mean estimation and learning mixtures of spherical gaussians
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DOI：
10.1145/3188745.3188758
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发表时间：
2017-11
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通讯作者：
Ilias Diakonikolas;D. Kane;Alistair Stewart
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文献类型：
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作者：
Ilias Diakonikolas;D. Kane;Alistair Stewart
我们研究列表可码头的（稳健）高斯平均估计和相关的与球形高斯的学习混合物的相关问题。 t中的点是00是常数，我们的算法在多项式时间内运行，并且达到误差o（α）。 （1/α））并达到误差o（log3/2（1/α）），几乎与我们建立的θ（log1/2（1/α）的信息理论最佳结合匹配）。查询（SQ）的下限表明，我们的算法的复杂性在质量上接近可能的水平。对于均匀k混合的原型情况，身份协方差高斯我们获得以下操作：对于任何> 0我们的算法以样品复杂性和运行时间（n，1/δ，（k/）1/）学习准确性δ内的未知参数，我们的算法对损坏的数据无关已知的多项式时间算法至少需要K1/4（K/δ）。 ，1/δ，klogk）。大多数要点被损坏的地方。
We study the problem of list-decodable (robust) Gaussian mean estimation and the related problem of learning mixtures of separated spherical Gaussians. In the former problem, we are given a set T of points in n with the promise that an α-fraction of points in T, where 00 is a constant, our algorithm runs in polynomial time and achieves error O(α). For d = Θ(log(1/α)), our algorithm runs in time (n/α)O(log(1/α)) and achieves error O(log3/2(1/α)), almost matching the information-theoretically optimal bound of Θ(log1/2(1/α)) that we establish. We also give a Statistical Query (SQ) lower bound suggesting that the complexity of our algorithm is qualitatively close to best possible. Learning Mixtures of Spherical Gaussians. We give a learning algorithm for mixtures of spherical Gaussians, with unknown spherical covariances, that succeeds under significantly weaker separation assumptions compared to prior work. For the prototypical case of a uniform k-mixture of identity covariance Gaussians we obtain the following: For any >0, if the pairwise separation between the means is at least Ω(k+√log(1/δ)), our algorithm learns the unknown parameters within accuracy δ with sample complexity and running time (n, 1/δ, (k/)1/). Moreover, our algorithm is robust to a small dimension-independent fraction of corrupted data. The previously best known polynomial time algorithm required separation at least k1/4 (k/δ). Finally, our algorithm works under separation of Õ(log3/2(k)+√log(1/δ)) with sample complexity and running time (n, 1/δ, klogk). This bound is close to the information-theoretically minimum separation of Ω(√logk). Our main technical contribution is a new technique, using degree-d multivariate polynomials, to remove outliers from high-dimensional datasets where the majority of the points are corrupted.