Source Identification for Mixtures of Product Distributions

TL;DR

This paper presents a new algorithm for identifying the original sources of a mixture of product distributions on binary variables, with explicit complexity bounds and improved efficiency over previous methods.

Contribution

It introduces the first explicit computational complexity bound for source identification in such mixtures, improving upon prior algorithms that only learned the mixture without source parameters.

Findings

Algorithm identifies source parameters from approximate moments.

Runs in $2^{O(k^2)} n^{O(k)}$ arithmetic operations.

Provides a quantitative analysis of source identifiability.

Abstract

We give an algorithm for source identification of a mixture of $k$ product distributions on $n$ bits. This is a fundamental problem in machine learning with many applications. Our algorithm identifies the source parameters of an identifiable mixture, given, as input, approximate values of multilinear moments (derived, for instance, from a sufficiently large sample), using $2^{O(k^2)} n^{O(k)}$ arithmetic operations. Our result is the first explicit bound on the computational complexity of source identification of such mixtures. The running time improves previous results by Feldman, O'Donnell, and Servedio (FOCS 2005) and Chen and Moitra (STOC 2019) that guaranteed only learning the mixture (without parametric identification of the source). Our analysis gives a quantitative version of a qualitative characterization of identifiable sources that is due to Tahmasebi, Motahari, and Maddah-Ali (ISIT 2018).