# Recovery of Joint Probability Distribution from one-way marginals: Low rank Tensors and Random Projections

arXiv: https://arxiv.org/abs/2103.11864 · HTML: https://mlsanity.com/paper/2103.11864/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 5 of its 22 references that are in the map.
- First posted on arXiv 22 Mar 2021.

## Builds on (most cited first, 5 of 5)

- [Tensor Decomposition for Signal Processing and Machine Learning](https://mlsanity.com/paper/1607.01668/) (2016)
- [Sliced Wasserstein Distance for Learning Gaussian Mixture Models](https://mlsanity.com/paper/1711.05376/) (2017)
- [On Identifiability of Nonnegative Matrix Factorization](https://mlsanity.com/paper/1709.00614/) (2017)
- [Tensors, Learning, and 'Kolmogorov Extension' for Finite-alphabet Random Vectors](https://mlsanity.com/paper/1712.00205/) (2017)
- [Recovering Joint Probability of Discrete Random Variables from Pairwise Marginals](https://mlsanity.com/paper/2006.16912/) (2020)

## Built on by (most cited first, 3 of 3)

- [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)
- [Beyond Internal Data: Constructing Complete Datasets for Fairness Testing](https://mlsanity.com/paper/2507.18561/) (2025)

## Abstract (arXiv)

Joint probability mass function (PMF) estimation is a fundamental machine learning problem. The number of free parameters scales exponentially with respect to the number of random variables. Hence, most work on nonparametric PMF estimation is based on some structural assumptions such as clique factorization adopted by probabilistic graphical models, imposition of low rank on the joint probability tensor and reconstruction from 3-way or 2-way marginals, etc. In the present work, we link random projections of data to the problem of PMF estimation using ideas from tomography. We integrate this idea with the idea of low-rank tensor decomposition to show that we can estimate the joint density from just one-way marginals in a transformed space. We provide a novel algorithm for recovering factors of the tensor from one-way marginals, test it across a variety of synthetic and real-world datasets, and also perform MAP inference on the estimated model for classification.