The goal of this paper is to show that a single robust estimator of the mean of a multivariate Gaussian distribution can enjoy five desirable properties. First, it is computationally tractable in the sense that it can be computed in a time which is at most polynomial in dimension, sample size and the logarithm of the inverse of the contamination rate. Second, it is equivariant by translations and orthogonal transformations. Third, it has a high breakdown point equal to 0.5, and a nearly-minimax-rate-breakdown point approximately equal to 0.28. Fourth, it is minimax rate optimal when data consist of independent observations corrupted by adversarially chosen outliers. Fifth, it is asymptotically optimal when the rate of contamination tends to zero. The estimator is obtained by an iterative reweighting approach. Each sample point is assigned a weight that is iteratively updated using a convex optimization problem. We also establish a dimension-free non-asymptotic risk bound for the expected error of the proposed estimator. It is the first of this kind results in the literature and involves only the effective rank of the covariance matrix.

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We consider the problem of finding the matching map between two sets of d-dimensional vectors from noisy observations, where the second set contains outliers.The matching map is then an injection, which can be consistently detected only if the vectors of the second set are well separated. The main result shows that, in the high-dimensional setting, a detection region of unknown injection may be characterized by the sets of vectors for which the inlier-inlier distance is of order at least d 1/4 and the inlier-outlier distance is of order at least d 1/2 . These rates are achieved using the matching minimizing the sum of logarithms of distances between matched pairs of points. We also prove lower bounds establishing optimality of these rates. Finally, we report the results of numerical experiments on both synthetic and real world data that illustrate our theoretical results and provide further insight into the properties of the algorithms studied in this work.

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We consider the problem of estimating the matching map between two sets of feature-vectors observed in a noisy environment and contaminated by outliers. It was already known in the literature that in the outlier-free setting, the least sum of squares (LSS) and the least sum of logarithms (LSL) are both minimax-rate-optimal. It has been recently proved that the optimality properties of the LSS continue to hold in the case the data sets contain outliers. In this work, we show that the same is true for the LSL as well. Therefore, LSL has the same desirable properties as the LSS, and, in addition, it is minimax-rate-optimal in the outlier-free setting with heteroscedastic noise.