# Tucker Diffusion Model for High-dimensional Tensor Generation

arXiv: https://arxiv.org/abs/2604.00481 · HTML: https://mlsanity.com/paper/2604.00481/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models.
- Builds on 46 of its 65 references that are in the map.
- First posted on arXiv 1 Apr 2026.

## Placed in

- Diffusion models / [Diffusion foundations and sampling](https://mlsanity.com/diffusion/diffusion-foundations-and-sampling/)

## Builds on (most cited first, 30 of 46)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [High-Resolution Image Synthesis with Latent Diffusion Models](https://mlsanity.com/paper/2112.10752/) (2021)
- [Denoising Diffusion Implicit Models](https://mlsanity.com/paper/2010.02502/) (2020)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Deep Unsupervised Learning using Nonequilibrium Thermodynamics](https://mlsanity.com/paper/1503.03585/) (2015)
- [Generative Adversarial Networks](https://mlsanity.com/paper/2203.00667/) (2022)
- [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019)
- [Elucidating the Design Space of Diffusion-Based Generative Models](https://mlsanity.com/paper/2206.00364/) (2022)
- [FiLM: Visual Reasoning with a General Conditioning Layer](https://mlsanity.com/paper/1709.07871/) (2017)
- [Cascaded Diffusion Models for High Fidelity Image Generation](https://mlsanity.com/paper/2106.15282/) (2021)
- [An Introduction to Variational Autoencoders](https://mlsanity.com/paper/1906.02691/) (2019)
- [TUDataset: A collection of benchmark datasets for learning with graphs](https://mlsanity.com/paper/2007.08663/) (2020)
- [MedMNIST v2 -- A large-scale lightweight benchmark for 2D and 3D biomedical image classification](https://mlsanity.com/paper/2110.14795/) (2021)
- [Spectrally-normalized margin bounds for neural networks](https://mlsanity.com/paper/1706.08498/) (2017)
- [Generative Adversarial Networks: An Overview](https://mlsanity.com/paper/1710.07035/) (2017)
- [Benign Overfitting in Linear Regression](https://mlsanity.com/paper/1906.11300/) (2019)
- [The Liver Tumor Segmentation Benchmark (LiTS)](https://mlsanity.com/paper/1901.04056/) (2019)
- [Sampling is as easy as learning the score: theory for diffusion models with minimal data assumptions](https://mlsanity.com/paper/2209.11215/) (2022)
- [Regularisation of Neural Networks by Enforcing Lipschitz Continuity](https://mlsanity.com/paper/1804.04368/) (2018)
- [Score Approximation, Estimation and Distribution Recovery of Diffusion Models on Low-Dimensional Data](https://mlsanity.com/paper/2302.07194/) (2023)
- [The Principles of Diffusion Models](https://mlsanity.com/paper/2510.21890/) (2025)
- [An Overview of Diffusion Models: Applications, Guided Generation, Statistical Rates and Optimization](https://mlsanity.com/paper/2404.07771/) (2024)
- [Subspace Diffusion Generative Models](https://mlsanity.com/paper/2205.01490/) (2022)
- [Unveil Conditional Diffusion Models with Classifier-free Guidance: A Sharp Statistical Theory](https://mlsanity.com/paper/2403.11968/) (2024)
- [Noisy Matrix Completion: Understanding Statistical Guarantees for Convex Relaxation via Nonconvex Optimization](https://mlsanity.com/paper/1902.07698/) (2019)
- [Tensor SVD: Statistical and Computational Limits](https://mlsanity.com/paper/1703.02724/) (2017)
- [Neural Network-Based Score Estimation in Diffusion Models: Optimization and Generalization](https://mlsanity.com/paper/2401.15604/) (2024)
- [Efficient Diffusion Models: A Comprehensive Survey from Principles to Practices](https://mlsanity.com/paper/2410.11795/) (2024)
- [On Statistical Rates and Provably Efficient Criteria of Latent Diffusion Transformers (DiTs)](https://mlsanity.com/paper/2407.01079/) (2024)
- [Spectral State Compression of Markov Processes](https://mlsanity.com/paper/1802.02920/) (2018)

## Built on by (most cited first, 2 of 2)

- [DiffATS: Diffusion in Aligned Tensor Space](https://mlsanity.com/paper/2605.09275/) (2026)
- [Conditional Tensor Diffusion: Distributional Counterfactual Learning and Inference](https://mlsanity.com/paper/2609.25924/) (2026)

## Abstract (arXiv)

Statistical inference on large-dimensional tensor data has been extensively studied in the literature and widely used in economics, biology, machine learning, and other fields, but how to generate a structured tensor with a target distribution is still a new problem. As profound AI generators, diffusion models have achieved remarkable success in learning complex distributions. However, their extension to generating multi-linear tensor-valued observations remains underexplored. In this work, we propose a novel Tucker diffusion model for learning high-dimensional tensor distributions. We show that the score function admits a structured decomposition under the low Tucker rank assumption, allowing it to be both accurately approximated and efficiently estimated using a carefully tailored tensor-shaped architecture named Tucker-Unet. Furthermore, the distribution of generated tensors, induced by the estimated score function, converges to the true data distribution at a rate depending on the maximum of tensor mode dimensions, thereby offering a clear theoretical advantage over the naive vectorized approach, which has a product dependence. Empirically, compared to existing approaches, the Tucker diffusion model demonstrates strong practical potential in synthetic and real-world tensor generation tasks, achieving comparable and sometimes even superior statistical performance with significantly reduced training and sampling costs.