# SPD-DDPM: Denoising Diffusion Probabilistic Models in the Symmetric Positive Definite Space

arXiv: https://arxiv.org/abs/2312.08200 · HTML: https://mlsanity.com/paper/2312.08200/

## Key facts

- Cited by 14 papers in the map (data through 14 Oct 2026).
- Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models.
- Builds on 20 of its 45 references that are in the map.
- First posted on arXiv 13 Dec 2023.

## Placed in

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

## Builds on (most cited first, 20 of 20)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [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)
- [Diffusion Models Beat GANs on Image Synthesis](https://mlsanity.com/paper/2105.05233/) (2021)
- [Improved Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2102.09672/) (2021)
- [GLIDE: Towards Photorealistic Image Generation and Editing with Text-Guided Diffusion Models](https://mlsanity.com/paper/2112.10741/) (2021)
- [DPM-Solver: A Fast ODE Solver for Diffusion Probabilistic Model Sampling in Around 10 Steps](https://mlsanity.com/paper/2206.00927/) (2022)
- [RePaint: Inpainting using Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2201.09865/) (2022)
- [ProlificDreamer: High-Fidelity and Diverse Text-to-3D Generation with Variational Score Distillation](https://mlsanity.com/paper/2305.16213/) (2023)
- [Zero-Shot Image Restoration Using Denoising Diffusion Null-Space Model](https://mlsanity.com/paper/2212.00490/) (2022)
- [DiffDock: Diffusion Steps, Twists, and Turns for Molecular Docking](https://mlsanity.com/paper/2210.01776/) (2022)
- [SE(3) diffusion model with application to protein backbone generation](https://mlsanity.com/paper/2302.02277/) (2023)
- [A Riemannian Network for SPD Matrix Learning](https://mlsanity.com/paper/1608.04233/) (2016)
- [Riemannian batch normalization for SPD neural networks](https://mlsanity.com/paper/1909.02414/) (2019)
- [Fr'echet Regression for Random Objects with Euclidean Predictors](https://mlsanity.com/paper/1608.03012/) (2016)
- [A neural network based on SPD manifold learning for skeleton-based hand gesture recognition](https://mlsanity.com/paper/1904.12970/) (2019)
- [GeomNet: A Neural Network Based on Riemannian Geometries of SPD Matrix Space and Cholesky Space for 3D Skeleton-Based Interaction Recognition](https://mlsanity.com/paper/2111.13089/) (2021)
- [Riemannian Local Mechanism for SPD Neural Networks](https://mlsanity.com/paper/2201.10145/) (2022)
- [Automatic Analysis of Facial Expressions Based on Deep Covariance Trajectories](https://mlsanity.com/paper/1810.11392/) (2018)

## Built on by (most cited first, 14 of 14)

- [Exploring Typographic Visual Prompts Injection Threats in Cross-Modality Generation Models](https://mlsanity.com/paper/2503.11519/) (2025)
- [Riemannian Flow Matching for Brain Connectivity Matrices via Pullback Geometry](https://mlsanity.com/paper/2505.18193/) (2025)
- [Not Just Text: Uncovering Vision Modality Typographic Threats in Image Generation Models](https://mlsanity.com/paper/2412.05538/) (2024)
- [Geometric Neural Network based on Phase Space for BCI-EEG decoding](https://mlsanity.com/paper/2403.05645/) (2024)
- [Fast and Stable Riemannian Metrics on SPD Manifolds via Cholesky Product Geometry](https://mlsanity.com/paper/2407.02607/) (2024)
- [Prior-Guided Residual Diffusion: Calibrated and Efficient Medical Image Segmentation](https://mlsanity.com/paper/2509.01330/) (2025)
- [A probabilistic view on Riemannian machine learning models for SPD matrices](https://mlsanity.com/paper/2505.02402/) (2025)
- [ReManNet: A Riemannian Manifold Network for Monocular 3D Lane Detection](https://mlsanity.com/paper/2603.19776/) (2026)
- [Marrying Generative Model of Healthcare Events with Digital Twin of Social Determinants of Health for Disease Reasoning](https://mlsanity.com/paper/2605.09771/) (2026)
- [Riemannian Diffusion Models on General Manifolds via Physics-Informed Neural Networks](https://mlsanity.com/paper/2605.31106/) (2026)
- [Deep Single-Index Fr'echet Regression](https://mlsanity.com/paper/2606.06957/) (2026)
- [The Reverse Telescoping Coordinate System for Positive Definite Matrices: Geometry, Computation, and Generative Modeling](https://mlsanity.com/paper/2606.15442/) (2026)
- [TVGL-CFM:Generating and Forecasting Time-Varying Trajectories of Dynamic Networks with Conditional Flow Matching](https://mlsanity.com/paper/2607.16894/) (2026)
- [CorrFlow: Exact and Permutation-Equivariant Flow Matching on the Correlation Elliptope](https://mlsanity.com/paper/2610.13606/) (2026)

## Abstract (arXiv)

Symmetric positive definite~(SPD) matrices have shown important value and applications in statistics and machine learning, such as FMRI analysis and traffic prediction. Previous works on SPD matrices mostly focus on discriminative models, where predictions are made directly on E(X|y), where y is a vector and X is an SPD matrix. However, these methods are challenging to handle for large-scale data, as they need to access and process the whole data. In this paper, inspired by denoising diffusion probabilistic model~(DDPM), we propose a novel generative model, termed SPD-DDPM, by introducing Gaussian distribution in the SPD space to estimate E(X|y). Moreover, our model is able to estimate p(X) unconditionally and flexibly without giving y. On the one hand, the model conditionally learns p(X|y) and utilizes the mean of samples to obtain E(X|y) as a prediction. On the other hand, the model unconditionally learns the probability distribution of the data p(X) and generates samples that conform to this distribution. Furthermore, we propose a new SPD net which is much deeper than the previous networks and allows for the inclusion of conditional factors. Experiment results on toy data and real taxi data demonstrate that our models effectively fit the data distribution both unconditionally and unconditionally and provide accurate predictions.