# Generative Modeling via Hierarchical Tensor Sketching

arXiv: https://arxiv.org/abs/2304.05305 · HTML: https://mlsanity.com/paper/2304.05305/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- Builds on 18 of its 53 references that are in the map.
- First posted on arXiv 11 Apr 2023.

## Builds on (most cited first, 18 of 18)

- [Auto-Encoding Variational Bayes](https://mlsanity.com/paper/1312.6114/) (2013)
- [Variational Inference with Normalizing Flows](https://mlsanity.com/paper/1505.05770/) (2015)
- [Density estimation using Real NVP](https://mlsanity.com/paper/1605.08803/) (2016)
- [Masked Autoregressive Flow for Density Estimation](https://mlsanity.com/paper/1705.07057/) (2017)
- [An Introduction to Matrix Concentration Inequalities](https://mlsanity.com/paper/1501.01571/) (2015)
- [MADE: Masked Autoencoder for Distribution Estimation](https://mlsanity.com/paper/1502.03509/) (2015)
- [Density Modeling of Images using a Generalized Normalization Transformation](https://mlsanity.com/paper/1511.06281/) (2015)
- [Neural Autoregressive Distribution Estimation](https://mlsanity.com/paper/1605.02226/) (2016)
- [Unsupervised Generative Modeling Using Matrix Product States](https://mlsanity.com/paper/1709.01662/) (2017)
- [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)
- [Generative modeling via tensor train sketching](https://mlsanity.com/paper/2202.11788/) (2022)
- [Scalable Gaussian Process Computations Using Hierarchical Matrices](https://mlsanity.com/paper/1808.03215/) (2018)
- [Generative Modeling via Tree Tensor Network States](https://mlsanity.com/paper/2209.01341/) (2022)
- [A Hierarchical Optimization Architecture for Large-Scale Power Networks](https://mlsanity.com/paper/2002.09796/) (2020)
- [Multiscale semidefinite programming approach to positioning problems with pairwise structure](https://mlsanity.com/paper/2012.10046/) (2020)
- [Error Analysis of Tensor-Train Cross Approximation](https://mlsanity.com/paper/2207.04327/) (2022)

## Built on by (most cited first, 8 of 8)

- [Ensemble-Based Annealed Importance Sampling](https://mlsanity.com/paper/2401.15645/) (2024)
- [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)
- [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)
- [Optimization-Free Diffusion Model -- A Perturbation Theory Approach](https://mlsanity.com/paper/2505.23652/) (2025)
- [Optimal Bias-variance Tradeoff in Matrix and Tensor Estimation](https://mlsanity.com/paper/2509.17382/) (2025)
- [On the Approximation Complexity of Matrix Product Operator (MPO) Born Machines](https://mlsanity.com/paper/2605.11471/) (2026)

## Abstract (arXiv)

We propose a hierarchical tensor-network approach for approximating high-dimensional probability density via empirical distribution. This leverages randomized singular value decomposition (SVD) techniques and involves solving linear equations for tensor cores in this tensor network. The complexity of the resulting algorithm scales linearly in the dimension of the high-dimensional density. An analysis of estimation error demonstrates the effectiveness of this method through several numerical experiments.