# Uniform Consistency in Nonparametric Mixture Models

arXiv: https://arxiv.org/abs/2108.14003 · HTML: https://mlsanity.com/paper/2108.14003/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 16 of its 103 references that are in the map.
- First posted on arXiv 31 Aug 2021.

## Builds on (most cited first, 16 of 16)

- [Statistical guarantees for the EM algorithm: From population to sample-based analysis](https://mlsanity.com/paper/1408.2156/) (2014)
- [Learning latent causal graphs via mixture oracles](https://mlsanity.com/paper/2106.15563/) (2021)
- [Convergence Rates for Gaussian Mixtures of Experts](https://mlsanity.com/paper/1907.04377/) (2019)
- [Learning Mixtures of Linear Regressions with Nearly Optimal Complexity](https://mlsanity.com/paper/1802.07895/) (2018)
- [On Learning Mixtures of Well-Separated Gaussians](https://mlsanity.com/paper/1710.11592/) (2017)
- [EM Converges for a Mixture of Many Linear Regressions](https://mlsanity.com/paper/1905.12106/) (2019)
- [On the Minimax Optimality of the EM Algorithm for Learning Two-Component Mixed Linear Regression](https://mlsanity.com/paper/2006.02601/) (2020)
- [On the nonparametric maximum likelihood estimator for Gaussian location mixture densities with application to Gaussian denoising](https://mlsanity.com/paper/1712.02009/) (2017)
- [A Universal Approximation Theorem for Mixture of Experts Models](https://mlsanity.com/paper/1602.03683/) (2016)
- [Identifiability of Nonparametric Mixture Models and Bayes Optimal Clustering](https://mlsanity.com/paper/1802.04397/) (2018)
- [Breaking the gridlock in Mixture-of-Experts: Consistent and Efficient Algorithms](https://mlsanity.com/paper/1802.07417/) (2018)
- [Deep Divergence-Based Approach to Clustering](https://mlsanity.com/paper/1902.04981/) (2019)
- [Singularity structures and impacts on parameter estimation in finite mixtures of distributions](https://mlsanity.com/paper/1609.02655/) (2016)
- [Nonparametric modal regression](https://mlsanity.com/paper/1412.1716/) (2014)
- [Consistent Estimation of Identifiable Nonparametric Mixture Models from Grouped Observations](https://mlsanity.com/paper/2006.07459/) (2020)
- [A convex program for bilinear inversion of sparse vectors](https://mlsanity.com/paper/1809.08359/) (2018)

## Built on by (most cited first, 5 of 5)

- [Tight Bounds on the Hardness of Learning Simple Nonparametric Mixtures](https://mlsanity.com/paper/2203.15150/) (2022)
- [Sample Complexity Using Infinite Multiview Models](https://mlsanity.com/paper/2302.04292/) (2023)
- [Generalized Identifiability Bounds for Mixture Models with Grouped Samples](https://mlsanity.com/paper/2207.11164/) (2022)
- [PMODE: Theoretically Grounded and Modular Mixture Modeling](https://mlsanity.com/paper/2508.21396/) (2025)
- [Model-free Estimation of Latent Structure via Multiscale Nonparametric Maximum Likelihood](https://mlsanity.com/paper/2410.22248/) (2024)

## Abstract (arXiv)

We study uniform consistency in nonparametric mixture models as well as closely related mixture of regression (also known as mixed regression) models, where the regression functions are allowed to be nonparametric and the error distributions are assumed to be convolutions of a Gaussian density. We construct uniformly consistent estimators under general conditions while simultaneously highlighting several pain points in extending existing pointwise consistency results to uniform results. The resulting analysis turns out to be nontrivial, and several novel technical tools are developed along the way. In the case of mixed regression, we prove L1 convergence of the regression functions while allowing for the component regression functions to intersect arbitrarily often, which presents additional technical challenges. We also consider generalizations to general (i.e. non-convolutional) nonparametric mixtures.