Model Selection and Parameter Estimation of One-Dimensional Gaussian Mixture Models

TL;DR

This paper investigates the sample complexity for learning one-dimensional Gaussian mixture models, establishing optimal bounds and proposing a Fourier-based method that achieves these bounds with improved efficiency and accuracy.

Contribution

It provides the first optimal sample complexity bounds for model order estimation in 1D GMMs and introduces a Fourier-based algorithm that attains these bounds.

Findings

Established fundamental lower bounds on sample complexity for GMM model order estimation.

Proposed a Fourier-based algorithm that matches the optimal sample complexity bounds.

Numerical experiments demonstrate superior efficiency and accuracy of the proposed method.

Abstract

In this paper, we study the problem of learning one-dimensional Gaussian mixture models (GMMs) with a specific focus on estimating both the model order and the mixing distribution from independent and identically distributed (i.i.d.) samples. This paper establishes the optimal sampling complexity for model order estimation in one-dimensional Gaussian mixture models. We prove a fundamental lower bound on the number of samples required to correctly identify the number of components with high probability, showing that this limit depends critically on the separation between component means and the total number of components. We then propose a Fourier-based approach to estimate both the model order and the mixing distribution. Our algorithm utilizes Fourier measurements constructed from the samples, and our analysis demonstrates that its sample complexity matches the established lower bound, thereby confirming its optimality. Numerical experiments further show that our method outperforms conventional techniques in terms of efficiency and accuracy.