Settling the Robust Learnability of Mixtures of Gaussians

TL;DR

This paper introduces a robust algorithm for learning mixtures of Gaussians with provable guarantees, leveraging differential operations on generating functions to achieve dimension-independent identifiability.

Contribution

It presents the first robust, provable method for learning any constant number of Gaussian mixtures under mild assumptions, using novel differential techniques and sum-of-squares analysis.

Findings

First robust algorithm for Gaussian mixtures with provable guarantees

Dimension-independent identifiability through differential operations

Effective sum-of-squares relaxation analysis

Abstract

This work represents a natural coalescence of two important lines of work: learning mixtures of Gaussians and algorithmic robust statistics. In particular we give the first provably robust algorithm for learning mixtures of any constant number of Gaussians. We require only mild assumptions on the mixing weights (bounded fractionality) and that the total variation distance between components is bounded away from zero. At the heart of our algorithm is a new method for proving dimension-independent polynomial identifiability through applying a carefully chosen sequence of differential operations to certain generating functions that not only encode the parameters we would like to learn but also the system of polynomial equations we would like to solve. We show how the symbolic identities we derive can be directly used to analyze a natural sum-of-squares relaxation.