# List-Decodable Mean Estimation in Nearly-PCA Time

Traditionally, robust statistics has focused on designing estimators tolerant to a minority of contaminated data. Robust list-decodable learning focuses on the more challenging regime where only a minority 1/k fraction of the dataset is drawn from the distribution of interest, and no assumptions are made on the remaining data. We study the fundamental task of list-decodable mean estimation in high dimensions. Our main result is a new list-decodable mean estimation algorithm for bounded covariance distributions with optimal sample complexity and error rate, running in nearly-PCA time. Assuming the ground truth distribution on ℝ^d has bounded covariance, our algorithm outputs a list of O(k) candidate means, one of which is within distance O(√(k)) from the truth. Our algorithm runs in time O(ndk) for all k = O(√(d)) ∪Ω(d), where n is the size of the dataset. We also show that a variant of our algorithm has runtime O(ndk) for all k, at the expense of an O(√(log k)) factor in the recovery guarantee. This runtime matches up to logarithmic factors the cost of performing a single k-PCA on the data, which is a natural bottleneck of known algorithms for (very) special cases of our problem, such as clustering well-separated mixtures. Prior to our work, the fastest list-decodable mean estimation algorithms had runtimes O(n^2 d k^2) and O(nd k^≥ 6). Our approach builds on a novel soft downweighting method, 𝖲𝖨𝖥𝖳, which is arguably the simplest known polynomial-time mean estimation technique in the list-decodable learning setting. To develop our fast algorithms, we boost the computational cost of 𝖲𝖨𝖥𝖳 via a careful "win-win-win" analysis of an approximate Ky Fan matrix multiplicative weights procedure we develop, which we believe may be of independent interest.

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