Convergence of latent mixing measures in finite and infinite mixture models (1109.3250v5)

Abstract: This paper studies convergence behavior of latent mixing measures that arise in finite and infinite mixture models, using transportation distances (i.e., Wasserstein metrics). The relationship between Wasserstein distances on the space of mixing measures and f-divergence functionals such as Hellinger and Kullback-Leibler distances on the space of mixture distributions is investigated in detail using various identifiability conditions. Convergence in Wasserstein metrics for discrete measures implies convergence of individual atoms that provide support for the measures, thereby providing a natural interpretation of convergence of clusters in clustering applications where mixture models are typically employed. Convergence rates of posterior distributions for latent mixing measures are established, for both finite mixtures of multivariate distributions and infinite mixtures based on the Dirichlet process.

• The paper establishes that convergence in the Wasserstein metric naturally reflects the convergence of latent mixing measures, yielding actionable contraction rates.
• The paper derives identifiability conditions that connect f-divergence between mixture densities with convergence bounds for mixing measures.
• The paper demonstrates practical implications for clustering and density estimation by providing precise posterior convergence rates in Bayesian nonparametric models.

Convergence of Latent Mixing Measures in Finite and Infinite Mixture Models

In the paper "Convergence of latent mixing measures in finite and infinite mixture models," the author, XuanLong Nguyen, examines the convergence behaviors of latent mixing measures in various mixture models. These models include finite mixtures and infinite mixtures such as those based on the Dirichlet process. The analytical focus is on the use of Wasserstein metrics, also known as transportation distances, to examine convergence properties.

The paper aims to establish a relationship between Wasserstein distances on the space of mixing measures and $f$-divergence functionals such as Hellinger and Kullback-Leibler (KL) distances on the space of mixture distributions. These relationships are elucidated under various identifiability conditions. The manuscript explores how the convergence in Wasserstein metrics for discrete measures translates to the convergence of atoms that support these measures, offering significant implications for clustering applications.

Key Contributions and Results

1. Wasserstein Metric and Convergence:
   • The paper establishes that convergence in the Wasserstein metric provides a natural interpretation of the convergence of clusters.
   • The use of Wasserstein metric allows for the analysis of posterior distributions' convergence rates for latent mixing measures, applicable to both finite mixtures of multivariate distributions and Dirichlet process-based infinite mixtures.
2. Identifiability and Convergence Conditions:
   • The work provides conditions under which the convergence of mixture densities entails convergence of mixing measures in Wasserstein metrics.
   • Theorems are presented that offer upper bounds on Wasserstein distances in terms of divergences between mixture densities based on identifiability conditions.
3. Posterior Convergence in Bayesian Context:
   • The paper focuses on the behavior of posterior distributions for Bayesian nonparametric mixture models.
   • Theorems are derived that establish contraction rates of posterior distributions for mixing measures, with conditions articulated purely in terms of mixing measures rather than mixture densities.
4. Specific Mixture Model Applications:
   • For finite mixtures with fixed support, the posterior convergence rate is $(\log n)^{1/4}n^{-1/4}$.
   • For Dirichlet process mixtures, rates are established based on smoothness conditions of likelihood densities. Specifically, ordinary smooth likelihoods achieve rates that scale with $(\log n/n)^{\gamma}$, whereas supersmooth likelihoods align with $(\log n)^{-1/\beta}$.

Implications

The research has both theoretical and practical implications. Theoretically, it expands the understanding of convergence behaviors in mixed models by leveraging the Wasserstein metric. Practically, the results could be highly beneficial for applications involving clustering and density estimation in high-dimensional and complex data contexts, where mixture models are typically employed. The establishment of convergence rates directly in terms of mixing measures offers new avenues in Bayesian nonparametric statistics, where the convergence of complex infinite-dimensional objects is of paramount interest.

The findings suggest potential future developments in the use of optimal transport methods within statistical models, particularly in enhancing the robustness and interpretability of clustering techniques in machine learning and AI. As computational techniques for Wasserstein distances continue to improve, these theoretical insights may pave the way for more efficient and effective algorithms in practice.

In conclusion, XuanLong Nguyen's paper encompasses an in-depth exploration of the convergence of mixing measures, underscored by substantial theoretical results with promising practical applications within the field of mixture models.