An Operator Theoretic Approach to Nonparametric Mixture Models

TL;DR

This paper introduces an operator-theoretic framework to analyze the identifiability of nonparametric mixture models with grouped data, providing conditions for recovery and new spectral algorithms for component estimation.

Contribution

It offers the first precise characterization of the sample size needed for identifiability in nonparametric mixture models without distributional assumptions, using an innovative operator-theoretic approach.

Findings

Identifiability depends on the number of observations per group and the number of mixture components.

Spectral algorithms can recover mixture components under the proposed framework.

The analysis applies to multinomial mixture models and demonstrates practical recovery on synthetic data.

Abstract

When estimating finite mixture models, it is common to make assumptions on the mixture components, such as parametric assumptions. In this work, we make no distributional assumptions on the mixture components and instead assume that observations from the mixture model are grouped, such that observations in the same group are known to be drawn from the same mixture component. We precisely characterize the number of observations $n$ per group needed for the mixture model to be identifiable, as a function of the number $m$ of mixture components. In addition to our assumption-free analysis, we also study the settings where the mixture components are either linearly independent or jointly irreducible. Furthermore, our analysis considers two kinds of identifiability -- where the mixture model is the simplest one explaining the data, and where it is the only one. As an application of these results, we precisely characterize identifiability of multinomial mixture models. Our analysis relies on an operator-theoretic framework that associates mixture models in the grouped-sample setting with certain infinite-dimensional tensors. Based on this framework, we introduce general spectral algorithms for recovering the mixture components and illustrate their use on a synthetic data set.