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.

本论文研究线性回归问题并提出了一种新的算法，它能够在存在离群值的情况下，对有限矩（至$L_4$）的样本进行最佳的次高斯误差边界估计，并且通过使用谱方法研究了线性回归问题与最远超平面问题之间的关系，同时引入了第三个经验过程进行统计学属性的研究。

使用亚高斯率的光谱算法进行稳健回归