# DiffATS: Diffusion in Aligned Tensor Space

arXiv: https://arxiv.org/abs/2605.09275 · HTML: https://mlsanity.com/paper/2605.09275/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models.
- Builds on 19 of its 66 references that are in the map.
- First posted on arXiv 10 May 2026.

## Placed in

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

## Builds on (most cited first, 19 of 19)

- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [DINOv2: Learning Robust Visual Features without Supervision](https://mlsanity.com/paper/2304.07193/) (2023)
- [SDXL: Improving Latent Diffusion Models for High-Resolution Image Synthesis](https://mlsanity.com/paper/2307.01952/) (2023)
- [Auto-Encoding Variational Bayes](https://mlsanity.com/paper/1312.6114/) (2013)
- [Progressive Growing of GANs for Improved Quality, Stability, and Variation](https://mlsanity.com/paper/1710.10196/) (2017)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [DiffWave: A Versatile Diffusion Model for Audio Synthesis](https://mlsanity.com/paper/2009.09761/) (2020)
- [Towards Accurate Generative Models of Video: A New Metric & Challenges](https://mlsanity.com/paper/1812.01717/) (2018)
- [AudioLDM: Text-to-Audio Generation with Latent Diffusion Models](https://mlsanity.com/paper/2301.12503/) (2023)
- [Latte: Latent Diffusion Transformer for Video Generation](https://mlsanity.com/paper/2401.03048/) (2024)
- [GeoDiff: a Geometric Diffusion Model for Molecular Conformation Generation](https://mlsanity.com/paper/2203.02923/) (2022)
- [Sora: A Review on Background, Technology, Limitations, and Opportunities of Large Vision Models](https://mlsanity.com/paper/2402.17177/) (2024)
- [Latent Video Diffusion Models for High-Fidelity Long Video Generation](https://mlsanity.com/paper/2211.13221/) (2022)
- [GenCast: Diffusion-based ensemble forecasting for medium-range weather](https://mlsanity.com/paper/2312.15796/) (2023)
- [From Zero to Turbulence: Generative Modeling for 3D Flow Simulation](https://mlsanity.com/paper/2306.01776/) (2023)
- [DCTdiff: Intriguing Properties of Image Generative Modeling in the DCT Space](https://mlsanity.com/paper/2412.15032/) (2024)
- [Generating Full-field Evolution of Physical Dynamics from Irregular Sparse Observations](https://mlsanity.com/paper/2505.09284/) (2025)
- [Rethinking Diffusion Models with Symmetries through Canonicalization with Applications to Molecular Graph Generation](https://mlsanity.com/paper/2602.15022/) (2026)
- [Tucker Diffusion Model for High-dimensional Tensor Generation](https://mlsanity.com/paper/2604.00481/) (2026)

## Abstract (arXiv)

Direct diffusion modeling of high-resolution spatiotemporal fields is computationally challenging. Parameter-efficient primitives address this by representing high-dimensional data with a compact set of parameters. In this paper, we construct data-dependent tensor primitives without pretrained compression autoencoders. Our construction starts from Tucker decomposition, which captures low-rank multilinear structure through a core tensor and mode-wise factors. However, Tucker factors are non-unique: the same tensor can be represented by different rotated factors, which complicates generative modeling. We address this issue with orthogonal Procrustes (OP) alignment. Specifically, we select medoid anchor matrices from the data and align the factor matrices to resolve the gauge ambiguity. This yields matrix Grassmannian primitives and tensor Grassmannian primitives that are compact, data-adaptive, and directly decodable by explicit multilinear reconstruction. Theoretically, we prove that the proposed primitive maps are homeomorphisms between low-rank tensors and their corresponding primitive spaces, certifying that the representations are non-degenerate and topologically faithful. Building on these primitives, we propose *Diffusion in Aligned Tensor Space* (DiffATS), a generative framework that trains diffusion models directly on aligned tensor primitives. Across images, videos, and PDE solutions, DiffATS achieves strong unconditional and conditional generation performance while compressing original data by 3.9× to 210×, without relying on any pretrained deep compression autoencoders.