Being Robust (in High Dimensions) Can Be Practical
This work addresses the problem of robust statistical estimation for high-dimensional data, making it practical for real-world applications, though it builds incrementally on prior theoretical advances.
The paper tackles the challenge of robust estimation in high dimensions, where existing methods are often intractable or tolerate few errors, by developing algorithms with optimal sample complexity and improved corruption tolerance, achieving state-of-the-art performance on synthetic and real data.
Robust estimation is much more challenging in high dimensions than it is in one dimension: Most techniques either lead to intractable optimization problems or estimators that can tolerate only a tiny fraction of errors. Recent work in theoretical computer science has shown that, in appropriate distributional models, it is possible to robustly estimate the mean and covariance with polynomial time algorithms that can tolerate a constant fraction of corruptions, independent of the dimension. However, the sample and time complexity of these algorithms is prohibitively large for high-dimensional applications. In this work, we address both of these issues by establishing sample complexity bounds that are optimal, up to logarithmic factors, as well as giving various refinements that allow the algorithms to tolerate a much larger fraction of corruptions. Finally, we show on both synthetic and real data that our algorithms have state-of-the-art performance and suddenly make high-dimensional robust estimation a realistic possibility.