Robust Mean Estimation on Highly Incomplete Data with Arbitrary Outliers

TL;DR

This paper presents algorithms for robustly estimating the mean of high-dimensional distributions with incomplete data and arbitrary outliers, achieving optimal error guarantees efficiently.

Contribution

It extends robust mean estimation methods to settings with highly incomplete data and outliers, providing nearly-linear time algorithms with optimal guarantees.

Findings

Achieves dimension-independent error bounds

Handles highly incomplete data with missing entries

Operates in nearly-linear time with respect to data size and dimension

Abstract

We study the problem of robustly estimating the mean of a $d$-dimensional distribution given $N$ examples, where most coordinates of every example may be missing and $\varepsilon N$ examples may be arbitrarily corrupted. Assuming each coordinate appears in a constant factor more than $\varepsilon N$ examples, we show algorithms that estimate the mean of the distribution with information-theoretically optimal dimension-independent error guarantees in nearly-linear time $\widetilde O(Nd)$. Our results extend recent work on computationally-efficient robust estimation to a more widely applicable incomplete-data setting.