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