We consider the following basic problem: Given corrupted samples from a high-dimensional Gaussian, can we efficiently learn its parameters? This is the prototypical question in robust statistics, a field that took shape in the 1960's with the pioneering works of Tukey and Huber. Unfortunately, all known robust estimators are hard to compute in high dimensions. This prompts the following question: Can we reconcile robustness and computational efficiency in high-dimensional learning?
In this talk, I will describe the first efficient algorithms for robustly learning a high-dimensional Gaussian that are able to tolerate a constant fraction of corruptions. More broadly, I will present a set of algorithmic techniques that yield efficient robust estimators for high-dimensional models under fairly general conditions.