# Characterizing Heterogeneous Rates in Finite Mixture Estimation via Partial Optimal Transport

arXiv: https://arxiv.org/abs/2609.16622 · HTML: https://mlsanity.com/paper/2609.16622/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 12 of its 48 references that are in the map.
- First posted on arXiv 15 Sep 2026.

## Builds on (most cited first, 12 of 12)

- [Variational Deep Embedding: An Unsupervised and Generative Approach to Clustering](https://mlsanity.com/paper/1611.05148/) (2016)
- [FuseMoE: Mixture-of-Experts Transformers for Fleximodal Fusion](https://mlsanity.com/paper/2402.03226/) (2024)
- [Mixture of Experts Meets Prompt-Based Continual Learning](https://mlsanity.com/paper/2405.14124/) (2024)
- [Convergence Rates for Gaussian Mixtures of Experts](https://mlsanity.com/paper/1907.04377/) (2019)
- [Deep Gaussian Mixture Models](https://mlsanity.com/paper/1711.06929/) (2017)
- [Refined Convergence Rates for Maximum Likelihood Estimation under Finite Mixture Models](https://mlsanity.com/paper/2202.08786/) (2022)
- [Optimal estimation of high-dimensional location Gaussian mixtures](https://mlsanity.com/paper/2002.05818/) (2020)
- [Model-based clustering based on sparse finite Gaussian mixtures](https://mlsanity.com/paper/1606.06828/) (2016)
- [On Multimarginal Partial Optimal Transport: Equivalent Forms and Computational Complexity](https://mlsanity.com/paper/2108.07992/) (2021)
- [Finite mixtures of matrix-variate Poisson-log normal distributions for three-way count data](https://mlsanity.com/paper/1807.08380/) (2018)
- [A transformation-based approach to Gaussian mixture density estimation for bounded data](https://mlsanity.com/paper/1807.06690/) (2018)
- [On the Geometry of Separation in Finite Gaussian Mixtures](https://mlsanity.com/paper/2606.16179/) (2026)

## Built on by (most cited first, 1 of 1)

- [Rarity Determines Minimax Rates of Semantic Entropy From Weak Pairwise Judgments With a Known Channel](https://mlsanity.com/paper/2610.17971/) (2026)

## Abstract (arXiv)

Parameter estimation in finite mixture models can exhibit highly heterogeneous convergence behavior: locally isolated components may be estimated substantially faster than groups of competing components. Existing analyses based on Wasserstein distances typically characterize only the worst-case rate and therefore do not fully capture this local heterogeneity. In this paper, we introduce a Voronoi-based partial optimal transport (VPOT) framework for obtaining refined local and global convergence guarantees for the maximum likelihood estimator of the mixing measure. The key geometric idea is to localize the comparison of two mixing measures to extended Voronoi neighborhoods and use partial optimal transport to accommodate the unequal masses of their local restrictions. Within each neighborhood, the first-order POT discrepancy is raised to a power determined by the number of locally competing atoms, allowing the resulting loss to adapt to the local degree of singularity. Under suitable regularity and strong identifiability conditions, we establish uniform local and global upper bounds for a maximum likelihood estimator under the VPOT loss. These bounds reveal a configuration-dependent form of parameter estimation: less singular local configurations admit faster convergence, whereas the most singular configuration recovers the classical worst-case behavior characterized by Wasserstein-based analyses. We further establish a minimax lower bound showing that the convergence rate for estimating the mixing measure under the VPOT loss is optimal. Our results hold in arbitrary fixed dimension without requiring mixing proportions to be uniformly bounded away from zero or prior knowledge of the true number of mixture components. Overall, VPOT provides a configuration-adaptive framework for capturing heterogeneous parameter-estimation behavior in finite mixture models.