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We study the problem of robust estimation of the mean vector of asub-Gaussian distribution. We introduce an estimator based on spectraldimension reduction (SDR) and establish a finite sample upper bound on itserror that is minimax-optimal up to a logarithmic factor. Furthermore, we provethat the breakdown point of the SDR estimator is equal to $1/2$, the highestpossible value of the breakdown point. In addition, the SDR estimator isequivariant by similarity transforms and has low computational complexity. Moreprecisely, in the case of $n$ vectors of dimension $p$ -- at most $\varepsilonn$ out of which are adversarially corrupted -- the SDR estimator has a squarederror of order $\big(\frac{r_\Sigma}{n} +\varepsilon^2\log(1/\varepsilon)\big){\log p}$ and a running time of order $p^3+ n p^2$. Here, $r_\Sigma\le p$ is the effective rank of the covariance matrixof the reference distribution. Another advantage of the SDR estimator is thatit does not require knowledge of the contamination rate and does not involvesample splitting. We also investigate extensions of the proposed algorithm andof the obtained results in the case of (partially) unknown covariance matrix.

Nearly minimax robust estimator of the mean vector by iterative spectral dimension reduction