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 frac 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 ℝd 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 widetildeO(ndk) for all k = O(√d) cup Omega(d), where n is the size of the dataset. We also show that a variant of our algorithm has runtime widetildeO(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 widetildeO(n2 d k2) and widetildeO(nd k≥ 6). Our approach builds on a novel soft downweighting method, mathsfSIFT, 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 mathsfSIFT 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.