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We study a new linear up to quadratic time algorithm for linear regression inthe absence of strong assumptions on the underlying distributions of samples,and in the presence of outliers. The goal is to design a procedure which comeswith actual working code that attains the optimal sub-gaussian error bound eventhough the data have only finite moments (up to $L_4$) and in the presence ofpossibly adversarial outliers. A polynomial-time solution to this problem hasbeen recently discovered but has high runtime due to its use of Sum-of-Squarehierarchy programming. At the core of our algorithm is an adaptation of thespectral method introduced for the mean estimation problem to the linearregression problem. As a by-product we established a connection between thelinear regression problem and the furthest hyperplane problem. From astochastic point of view, in addition to the study of the classical quadraticand multiplier processes we introduce a third empirical process that comesnaturally in the study of the statistical properties of the algorithm.

A spectral algorithm for robust regression with subgaussian rates