Modern data sets are vulnerable to outliers and rare events, making classical statistical methods, like mean estimation under IID assumptions, unreliable. This issue spurred the development of Robust Statistics, which introduced resilient estimators such as medians and trimmed means. However, many robust methods, like the Tukey median, are computationally infeasible in high dimensions. Between 2010 and 2015, progress was made with the development of new statistical guarantees and computational techniques, notably by Diakonikolas et al., which led to the field of Algorithmic Robust Statistics. Despite this, practical challenges remain: most methods require knowledge of the population covariance, rely on unknown distributional properties, and cannot robustly estimate covariance matrices in spectral norm—possibly an inherently hard problem. We revisit this problem with a novel lens based on notions from optimization, showing some advancement on these limitations via a precise “optimization landscape” for the robust mean and covariance estimation problem—specifically, an inexact Polyak–Łojasiewicz property. Finally, we present open questions and directions for future research, both in theory and in practical applications.

Texto informado pelo autor.

* Os participantes dos seminários não poderão acessar às dependências da FGV usando bermuda, chinelos, blusa modelo top ou cropped, minissaia ou camiseta regata. O uso da máscara é facultativo, porém é obrigatória a apresentação do comprovante de vacinação (físico ou digital).