# Multi-Subspace Multi-Modal Modeling for Diffusion Models: Estimation, Convergence and Mixture of Experts

arXiv: https://arxiv.org/abs/2601.01475 · HTML: https://mlsanity.com/paper/2601.01475/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models.
- Builds on 34 of its 40 references that are in the map.
- First posted on arXiv 4 Jan 2026.

## Placed in

- Diffusion models / [Diffusion foundations and sampling](https://mlsanity.com/diffusion/diffusion-foundations-and-sampling/)
- Mixture-of-experts / [MoE theory and gating functions](https://mlsanity.com/moe/moe-theory-and-gating-functions/)

## Builds on (most cited first, 30 of 34)

- [High-Resolution Image Synthesis with Latent Diffusion Models](https://mlsanity.com/paper/2112.10752/) (2021)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Video Diffusion Models](https://mlsanity.com/paper/2204.03458/) (2022)
- [Sampling is as easy as learning the score: theory for diffusion models with minimal data assumptions](https://mlsanity.com/paper/2209.11215/) (2022)
- [The Intrinsic Dimension of Images and Its Impact on Learning](https://mlsanity.com/paper/2104.08894/) (2021)
- [One-2-3-45++: Fast Single Image to 3D Objects with Consistent Multi-View Generation and 3D Diffusion](https://mlsanity.com/paper/2311.07885/) (2023)
- [Convergence of denoising diffusion models under the manifold hypothesis](https://mlsanity.com/paper/2208.05314/) (2022)
- [Deep Compression Autoencoder for Efficient High-Resolution Diffusion Models](https://mlsanity.com/paper/2410.10733/) (2024)
- [Score Approximation, Estimation and Distribution Recovery of Diffusion Models on Low-Dimensional Data](https://mlsanity.com/paper/2302.07194/) (2023)
- [Diffusion Models are Minimax Optimal Distribution Estimators](https://mlsanity.com/paper/2303.01861/) (2023)
- [EDTalk: Efficient Disentanglement for Emotional Talking Head Synthesis](https://mlsanity.com/paper/2404.01647/) (2024)
- [On the Generalization Properties of Diffusion Models](https://mlsanity.com/paper/2311.01797/) (2023)
- [Verifying the Union of Manifolds Hypothesis for Image Data](https://mlsanity.com/paper/2207.02862/) (2022)
- [Unveil Conditional Diffusion Models with Classifier-free Guidance: A Sharp Statistical Theory](https://mlsanity.com/paper/2403.11968/) (2024)
- [Gradient Guidance for Diffusion Models: An Optimization Perspective](https://mlsanity.com/paper/2404.14743/) (2024)
- [On the Intrinsic Dimensionality of Image Representations](https://mlsanity.com/paper/1803.09672/) (2018)
- [Learning Mixtures of Gaussians Using the DDPM Objective](https://mlsanity.com/paper/2307.01178/) (2023)
- [Breaking the Curse of Dimensionality: Diffusion Models Efficiently Learn Low-Dimensional Distributions](https://mlsanity.com/paper/2409.02426/) (2024)
- [Follow-Your-Click: Open-domain Regional Image Animation via Short Prompts](https://mlsanity.com/paper/2403.08268/) (2024)
- [Reward-Directed Conditional Diffusion: Provable Distribution Estimation and Reward Improvement](https://mlsanity.com/paper/2307.07055/) (2023)
- [Learning general Gaussian mixtures with efficient score matching](https://mlsanity.com/paper/2404.18893/) (2024)
- [Neural Network-Based Score Estimation in Diffusion Models: Optimization and Generalization](https://mlsanity.com/paper/2401.15604/) (2024)
- [A Geometric View of Data Complexity: Efficient Local Intrinsic Dimension Estimation with Diffusion Models](https://mlsanity.com/paper/2406.03537/) (2024)
- [On Statistical Rates and Provably Efficient Criteria of Latent Diffusion Transformers (DiTs)](https://mlsanity.com/paper/2407.01079/) (2024)
- [Deep Networks as Denoising Algorithms: Sample-Efficient Learning of Diffusion Models in High-Dimensional Graphical Models](https://mlsanity.com/paper/2309.11420/) (2023)
- [Analysis of learning a flow-based generative model from limited sample complexity](https://mlsanity.com/paper/2310.03575/) (2023)
- [On diffusion-based generative models and their error bounds: The log-concave case with full convergence estimates](https://mlsanity.com/paper/2311.13584/) (2023)
- [On Statistical Rates of Conditional Diffusion Transformers: Approximation, Estimation and Minimax Optimality](https://mlsanity.com/paper/2411.17522/) (2024)
- [High-dimensional Asymptotics of Denoising Autoencoders](https://mlsanity.com/paper/2305.11041/) (2023)
- [Understanding Representation Dynamics of Diffusion Models via Low-Dimensional Modeling](https://mlsanity.com/paper/2502.05743/) (2025)

## Built on by (most cited first, 1 of 1)

- [Recovering Weighted Tangent Geometry from a Single-Scale Score Field](https://mlsanity.com/paper/2608.22334/) (2026)

## Abstract (arXiv)

Recently, diffusion models have achieved a great performance with a small dataset of size n and a fast optimization process. However, the estimation error of diffusion models suffers from the curse of dimensionality n-1/D with the data dimension D. Since images are usually a union of low-dimensional manifolds, current works model the data as a union of linear subspaces with Gaussian latent and achieve a 1/√n bound. Though this modeling reflects the multi-manifold property, the Gaussian latent can not capture the multi-modal property of the latent manifold. To bridge this gap, we propose the mixture subspace of low-rank mixture of Gaussian (MoLR-MoG) modeling, which models the target data as a union of K linear subspaces, and each subspace admits a mixture of Gaussian latent (nk modals with dimension dk). With this modeling, the corresponding score function naturally has a mixture of expert (MoE) structure, captures the multi-modal information, and contains nonlinear property. We first conduct real-world experiments to show that the generation results of MoE-latent MoG NN are much better than MoE-latent Gaussian score. Furthermore, MoE-latent MoG NN achieves a comparable performance with MoE-latent Unet with 10 × parameters. These results indicate that the MoLR-MoG modeling is reasonable and suitable for real-world data. After that, based on such MoE-latent MoG score, we provide a R4√Sigmak=1Knk√Sigmak=1Knkdk/√n estimation error, which escapes the curse of dimensionality by using data structure. Finally, we study the optimization process and prove the convergence guarantee under the MoLR-MoG modeling. Combined with these results, under a setting close to real-world data, this work explains why diffusion models only require a small training sample and enjoy a fast optimization process to achieve a great performance.