# Generalized Identifiability Bounds for Mixture Models with Grouped Samples

arXiv: https://arxiv.org/abs/2207.11164 · HTML: https://mlsanity.com/paper/2207.11164/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 23 of its 71 references that are in the map.
- First posted on arXiv 22 Jul 2022.

## Builds on (most cited first, 23 of 23)

- [Efficient Estimation of Word Representations in Vector Space](https://mlsanity.com/paper/1301.3781/) (2013)
- [VAE with a VampPrior](https://mlsanity.com/paper/1705.07120/) (2017)
- [Deep Unsupervised Clustering with Gaussian Mixture Variational Autoencoders](https://mlsanity.com/paper/1611.02648/) (2016)
- [Variational Deep Embedding: An Unsupervised and Generative Approach to Clustering](https://mlsanity.com/paper/1611.05148/) (2016)
- [Composing graphical models with neural networks for structured representations and fast inference](https://mlsanity.com/paper/1603.06277/) (2016)
- [Identifiability of deep generative models without auxiliary information](https://mlsanity.com/paper/2206.10044/) (2022)
- [Learning Theory for Distribution Regression](https://mlsanity.com/paper/1411.2066/) (2014)
- [Multi-Facet Clustering Variational Autoencoders](https://mlsanity.com/paper/2106.05241/) (2021)
- [Distribution-Free Distribution Regression](https://mlsanity.com/paper/1302.0082/) (2013)
- [The More, the Merrier: the Blessing of Dimensionality for Learning Large Gaussian Mixtures](https://mlsanity.com/paper/1311.2891/) (2013)
- [Tensors, Learning, and 'Kolmogorov Extension' for Finite-alphabet Random Vectors](https://mlsanity.com/paper/1712.00205/) (2017)
- [Identifiability of Nonparametric Mixture Models and Bayes Optimal Clustering](https://mlsanity.com/paper/1802.04397/) (2018)
- [Low-rank Characteristic Tensor Density Estimation Part I: Foundations](https://mlsanity.com/paper/2008.12315/) (2020)
- [Learning Mixtures of Smooth Product Distributions: Identifiability and Algorithm](https://mlsanity.com/paper/1904.01156/) (2019)
- [One-Class Support Measure Machines for Group Anomaly Detection](https://mlsanity.com/paper/1303.0309/) (2013)
- [Maximum Likelihood Estimation for Learning Populations of Parameters](https://mlsanity.com/paper/1902.04553/) (2019)
- [Consistent Estimation of Identifiable Nonparametric Mixture Models from Grouped Observations](https://mlsanity.com/paper/2006.07459/) (2020)
- [Beyond Smoothness: Incorporating Low-Rank Analysis into Nonparametric Density Estimation](https://mlsanity.com/paper/2204.00930/) (2022)
- [Nonparametric Estimation of Multi-View Latent Variable Models](https://mlsanity.com/paper/1311.3287/) (2013)
- [Recovery Guarantees for Kernel-based Clustering under Non-parametric Mixture Models](https://mlsanity.com/paper/2110.09476/) (2021)
- [Uniform Consistency in Nonparametric Mixture Models](https://mlsanity.com/paper/2108.14003/) (2021)
- [Improving Nonparametric Density Estimation with Tensor Decompositions](https://mlsanity.com/paper/2010.02425/) (2020)
- [Tight Bounds on the Hardness of Learning Simple Nonparametric Mixtures](https://mlsanity.com/paper/2203.15150/) (2022)

## Built on by (most cited first, 2 of 2)

- [Sample Complexity Using Infinite Multiview Models](https://mlsanity.com/paper/2302.04292/) (2023)
- [PMODE: Theoretically Grounded and Modular Mixture Modeling](https://mlsanity.com/paper/2508.21396/) (2025)

## Abstract (arXiv)

Recent work has shown that finite mixture models with m components are identifiable, while making no assumptions on the mixture components, so long as one has access to groups of samples of size 2m-1 which are known to come from the same mixture component. In this work we generalize that result and show that, if every subset of k mixture components of a mixture model are linearly independent, then that mixture model is identifiable with only (2m-1)/(k-1) samples per group. We further show that this value cannot be improved. We prove an analogous result for a stronger form of identifiability known as "determinedness" along with a corresponding lower bound. This independence assumption almost surely holds if mixture components are chosen randomly from a k-dimensional space. We describe some implications of our results for multinomial mixture models and topic modeling.