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We study the problem of {\em list-decodable mean estimation} for boundedcovariance distributions. Specifically, we are given a set $T$ of points in$\mathbb{R}^d$ with the promise that an unknown $\alpha$-fraction of points in$T$, where $0< \alpha < 1/2$, are drawn from an unknown mean and boundedcovariance distribution $D$, and no assumptions are made on the remainingpoints. The goal is to output a small list of hypothesis vectors such that atleast one of them is close to the mean of $D$. We give the first practicallyviable estimator for this problem. In more detail, our algorithm is sample andcomputationally efficient, and achieves information-theoretically near-optimalerror. While the only prior algorithm for this setting inherently relied on theellipsoid method, our algorithm is iterative and only uses spectral techniques.Our main technical innovation is the design of a soft outlier removal procedurefor high-dimensional heavy-tailed datasets with a majority of outliers.

List-Decodable Mean Estimation via Iterative Multi-Filtering