## The Annals of Statistics

- Ann. Statist.
- Volume 47, Number 5 (2019), 2704-2733.

### An operator theoretic approach to nonparametric mixture models

Robert A. Vandermeulen and Clayton D. Scott

#### 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 a general spectral algorithm for recovering the mixture components.

#### Article information

**Source**

Ann. Statist., Volume 47, Number 5 (2019), 2704-2733.

**Dates**

Received: October 2016

Revised: March 2018

First available in Project Euclid: 3 August 2019

**Permanent link to this document**

https://projecteuclid.org/euclid.aos/1564797861

**Digital Object Identifier**

doi:10.1214/18-AOS1762

**Mathematical Reviews number (MathSciNet)**

MR3988770

**Subjects**

Primary: 62E10: Characterization and structure theory

Secondary: 62G05: Estimation

**Keywords**

Mixture model nonparametric mixture identifiability tensor factorization multinomial mixture topic model joint irreducibility

#### Citation

Vandermeulen, Robert A.; Scott, Clayton D. An operator theoretic approach to nonparametric mixture models. Ann. Statist. 47 (2019), no. 5, 2704--2733. doi:10.1214/18-AOS1762. https://projecteuclid.org/euclid.aos/1564797861

#### Supplemental materials

- Supplement to “An operator theoretic approach to nonparametric mixture models”. Technical results and additional algorithmic details.Digital Object Identifier: doi:10.1214/18-AOS1762SUPPSupplemental files are immediately available to subscribers. Non-subscribers gain access to supplemental files with the purchase of the article.