Robust Statistics, Revisited

Starting from the seminal works of Tukey (1960) and Huber (1964), the field of robust statistics asks: Are there estimators that provably work in the presence of noise? The trouble is that all known provably robust estimators are also hard to compute in high-dimensions.
Here, we study a basic problem in robust statistics, posed in various forms in the above works. Given corrupted samples from a high-dimensional Gaussian, are there efficient algorithms to accurately estimate its parameters? We give the first algorithms that are able to tolerate a constant fraction of corruptions that is independent of the dimension. Additionally, we give several more applications of our techniques to product distributions and various mixture models.
This is based on joint work with Ilias Diakonikolas, Jerry Li, Gautam Kamath, Daniel Kane and Alistair Stewart.