# Learning Multivariate CDFs and Copulas using Tensor Factorization

arXiv: https://arxiv.org/abs/2210.07132 · HTML: https://mlsanity.com/paper/2210.07132/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 24 of its 43 references that are in the map.
- First posted on arXiv 13 Oct 2022.

## Builds on (most cited first, 24 of 24)

- [A Style-Based Generator Architecture for Generative Adversarial Networks](https://mlsanity.com/paper/1812.04948/) (2018)
- [Generative Adversarial Networks](https://mlsanity.com/paper/2203.00667/) (2022)
- [Large Scale GAN Training for High Fidelity Natural Image Synthesis](https://mlsanity.com/paper/1809.11096/) (2018)
- [Density estimation using Real NVP](https://mlsanity.com/paper/1605.08803/) (2016)
- [WaveNet: A Generative Model for Raw Audio](https://mlsanity.com/paper/1609.03499/) (2016)
- [Glow: Generative Flow with Invertible 1x1 Convolutions](https://mlsanity.com/paper/1807.03039/) (2018)
- [Normalizing Flows for Probabilistic Modeling and Inference](https://mlsanity.com/paper/1912.02762/) (2019)
- [Masked Autoregressive Flow for Density Estimation](https://mlsanity.com/paper/1705.07057/) (2017)
- [FFJORD: Free-form Continuous Dynamics for Scalable Reversible Generative Models](https://mlsanity.com/paper/1810.01367/) (2018)
- [PixelCNN++: Improving the PixelCNN with Discretized Logistic Mixture Likelihood and Other Modifications](https://mlsanity.com/paper/1701.05517/) (2017)
- [MADE: Masked Autoencoder for Distribution Estimation](https://mlsanity.com/paper/1502.03509/) (2015)
- [Count-Based Exploration with Neural Density Models](https://mlsanity.com/paper/1703.01310/) (2017)
- [Tensor Decomposition for Signal Processing and Machine Learning](https://mlsanity.com/paper/1607.01668/) (2016)
- [Neural Autoregressive Flows](https://mlsanity.com/paper/1804.00779/) (2018)
- [Semi-Supervised Learning with Generative Adversarial Networks](https://mlsanity.com/paper/1606.01583/) (2016)
- [AdvFlow: Inconspicuous Black-box Adversarial Attacks using Normalizing Flows](https://mlsanity.com/paper/2007.07435/) (2020)
- [A Flexible and Efficient Algorithmic Framework for Constrained Matrix and Tensor Factorization](https://mlsanity.com/paper/1506.04209/) (2015)
- [Learning Latent Representations for Speech Generation and Transformation](https://mlsanity.com/paper/1704.04222/) (2017)
- [Neural Likelihoods via Cumulative Distribution Functions](https://mlsanity.com/paper/1811.00974/) (2018)
- [Tensors, Learning, and 'Kolmogorov Extension' for Finite-alphabet Random Vectors](https://mlsanity.com/paper/1712.00205/) (2017)
- [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)
- [Low-rank Characteristic Tensor Density Estimation Part II: Compression and Latent Density Estimation](https://mlsanity.com/paper/2106.10591/) (2021)
- [Nonparametric Estimation of Multi-View Latent Variable Models](https://mlsanity.com/paper/1311.3287/) (2013)

## Abstract (arXiv)

Learning the multivariate distribution of data is a core challenge in statistics and machine learning. Traditional methods aim for the probability density function (PDF) and are limited by the curse of dimensionality. Modern neural methods are mostly based on black-box models, lacking identifiability guarantees. In this work, we aim to learn multivariate cumulative distribution functions (CDFs), as they can handle mixed random variables, allow efficient box probability evaluation, and have the potential to overcome local sample scarcity owing to their cumulative nature. We show that any grid sampled version of a joint CDF of mixed random variables admits a universal representation as a naive Bayes model via the Canonical Polyadic (tensor-rank) decomposition. By introducing a low-rank model, either directly in the raw data domain, or indirectly in a transformed (Copula) domain, the resulting model affords efficient sampling, closed form inference and uncertainty quantification, and comes with uniqueness guarantees under relatively mild conditions. We demonstrate the superior performance of the proposed model in several synthetic and real datasets and applications including regression, sampling and data imputation. Interestingly, our experiments with real data show that it is possible to obtain better density/mass estimates indirectly via a low-rank CDF model, than a low-rank PDF/PMF model.