# Learning Mixtures of Smooth Product Distributions: Identifiability and Algorithm

arXiv: https://arxiv.org/abs/1904.01156 · HTML: https://mlsanity.com/paper/1904.01156/

## Key facts

- Cited by 14 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 2 Apr 2019.

## Built on by (most cited first, 14 of 14)

- [Tensor-Train Density Estimation](https://mlsanity.com/paper/2108.00089/) (2021)
- [Beyond Smoothness: Incorporating Low-Rank Analysis into Nonparametric Density Estimation](https://mlsanity.com/paper/2204.00930/) (2022)
- [Stochastic Mirror Descent for Low-Rank Tensor Decomposition Under Non-Euclidean Losses](https://mlsanity.com/paper/2104.14562/) (2021)
- [Moment Estimation for Nonparametric Mixture Models Through Implicit Tensor Decomposition](https://mlsanity.com/paper/2210.14386/) (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)
- [Generalization Bounds for Inductive Matrix Completion in Low-noise Settings](https://mlsanity.com/paper/2212.08339/) (2022)
- [Joint Probability Estimation Using Tensor Decomposition and Dictionaries](https://mlsanity.com/paper/2203.01667/) (2022)
- [Estimating Joint Probability Distribution With Low-Rank Tensor Decomposition, Radon Transforms and Dictionaries](https://mlsanity.com/paper/2304.08740/) (2023)
- [A Nested Matrix-Tensor Model for Noisy Multi-view Clustering](https://mlsanity.com/paper/2305.19992/) (2023)
- [PMODE: Theoretically Grounded and Modular Mixture Modeling](https://mlsanity.com/paper/2508.21396/) (2025)
- [Learning Multivariate CDFs and Copulas using Tensor Factorization](https://mlsanity.com/paper/2210.07132/) (2022)
- [Joint Bayesian Parameter and Model Order Estimation for Low-Rank Probability Mass Tensors](https://mlsanity.com/paper/2410.06329/) (2024)
- [A Unified MDL-based Binning and Tensor Factorization Framework for PDF Estimation](https://mlsanity.com/paper/2504.18686/) (2025)

## Abstract (arXiv)

We study the problem of learning a mixture model of non-parametric product distributions. The problem of learning a mixture model is that of finding the component distributions along with the mixing weights using observed samples generated from the mixture. The problem is well-studied in the parametric setting, i.e., when the component distributions are members of a parametric family -- such as Gaussian distributions. In this work, we focus on multivariate mixtures of non-parametric product distributions and propose a two-stage approach which recovers the component distributions of the mixture under a smoothness condition. Our approach builds upon the identifiability properties of the canonical polyadic (low-rank) decomposition of tensors, in tandem with Fourier and Shannon-Nyquist sampling staples from signal processing. We demonstrate the effectiveness of the approach on synthetic and real datasets.