# Generative modeling via tensor train sketching

arXiv: https://arxiv.org/abs/2202.11788 · HTML: https://mlsanity.com/paper/2202.11788/

## Key facts

- Cited by 13 papers in the map (data through 14 Oct 2026).
- Builds on 10 of its 36 references that are in the map.
- First posted on arXiv 23 Feb 2022.

## Builds on (most cited first, 10 of 10)

- [Auto-Encoding Variational Bayes](https://mlsanity.com/paper/1312.6114/) (2013)
- [Variational Inference with Normalizing Flows](https://mlsanity.com/paper/1505.05770/) (2015)
- [Spectral Methods for Data Science: A Statistical Perspective](https://mlsanity.com/paper/2012.08496/) (2020)
- [An Introduction to Deep Generative Modeling](https://mlsanity.com/paper/2103.05180/) (2021)
- [Unsupervised Generative Modeling Using Matrix Product States](https://mlsanity.com/paper/1709.01662/) (2017)
- [Tree Tensor Networks for Generative Modeling](https://mlsanity.com/paper/1901.02217/) (2019)
- [Efficient Low Rank Tensor Ring Completion](https://mlsanity.com/paper/1707.08184/) (2017)
- [Tensor-Train Density Estimation](https://mlsanity.com/paper/2108.00089/) (2021)
- [Modeling Sequences with Quantum States: A Look Under the Hood](https://mlsanity.com/paper/1910.07425/) (2019)
- [Low-Rank Tucker Approximation of a Tensor From Streaming Data](https://mlsanity.com/paper/1904.10951/) (2019)

## Built on by (most cited first, 13 of 13)

- [Generative Learning of Continuous Data by Tensor Networks](https://mlsanity.com/paper/2310.20498/) (2023)
- [Generative Modeling via Hierarchical Tensor Sketching](https://mlsanity.com/paper/2304.05305/) (2023)
- [Generative Modeling via Tree Tensor Network States](https://mlsanity.com/paper/2209.01341/) (2022)
- [High-dimensional density estimation with tensorizing flow](https://mlsanity.com/paper/2212.00759/) (2022)
- [Generalization and Overfitting in Matrix Product State Machine Learning Architectures](https://mlsanity.com/paper/2208.04372/) (2022)
- [Tensorization of neural networks for improved privacy and interpretability](https://mlsanity.com/paper/2501.06300/) (2025)
- [Plastic tensor networks for interpretable generative modeling](https://mlsanity.com/paper/2504.06722/) (2025)
- [Tensorizing flows: a tool for variational inference](https://mlsanity.com/paper/2305.02460/) (2023)
- [Tensor Network Estimation of Distribution Algorithms](https://mlsanity.com/paper/2412.19780/) (2024)
- [Combining Monte Carlo and Tensor-network Methods for Partial Differential Equations via Sketching](https://mlsanity.com/paper/2305.17884/) (2023)
- [Explaining Anomalies with Tensor Networks](https://mlsanity.com/paper/2505.03911/) (2025)
- [Optimal Bias-variance Tradeoff in Matrix and Tensor Estimation](https://mlsanity.com/paper/2509.17382/) (2025)
- [Private and interpretable clinical prediction with quantum-inspired tensor train models](https://mlsanity.com/paper/2602.06110/) (2026)

## Abstract (arXiv)

In this paper, we introduce a sketching algorithm for constructing a tensor train representation of a probability density from its samples. Our method deviates from the standard recursive SVD-based procedure for constructing a tensor train. Instead, we formulate and solve a sequence of small linear systems for the individual tensor train cores. This approach can avoid the curse of dimensionality that threatens both the algorithmic and sample complexities of the recovery problem. Specifically, for Markov models under natural conditions, we prove that the tensor cores can be recovered with a sample complexity that scales logarithmically in the dimensionality. Finally, we illustrate the performance of the method with several numerical experiments.