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SPD-DDPM: Denoising Diffusion Probabilistic Models in the Symmetric Positive Definite Space
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Proposes SPD-DDPM, a diffusion model that defines forward/backward Gaussian processes on symmetric positive definite matrices, with a deeper SPD U-Net, for unconditional SPD distribution fitting and conditional E(X|y) prediction on toy data and NYC taxi flows.
Problem
Estimating distributions and conditional expectations E(X|y) of symmetric positive definite (SPD) matrix-valued outputs; existing SPD methods are discriminative, struggle with large-scale data, and lack density estimation on the SPD manifold.
Approach
Extends DDPM to the SPD space by defining Gaussian distributions, a matrix-power/exp-log based addition and scalar multiplication (affine-invariant geometry), and derives forward and reverse processes; uses a deeper SPD U-Net with time/condition embeddings, and for conditional generation takes the Riemannian (Fréchet) mean of generated samples as the prediction of E(X|y).
Key results
- Unconditional: SPD-DDPM mean distance 23.17±3.00 vs DDPM 813.75±187.92 on toy SPD data.
- Conditional taxi data: SPD-DDPM Frobenius 6.4684 and affine-invariant 0.6123 vs Frechet Regression 6.6448 and 8.0099.
- Ablation: gamma=10 gives 23.17 mean distance vs 65.81 at gamma=2; removing down/up convolutions raises distance to 42.75.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use SPD-DDPM instead of DDPM | Unconditional generation · Toy data | Mean distance: 813.75$\pm$ 187.92 -> 23.17$\pm$ 3.00 | Helped |
| Use conditional SPD-DDPM instead of Frechet Regression | Conditional prediction · NYC taxi data | Frobenius: 6.6448 -> 6.4684 | Helped |
| Use conditional SPD-DDPM instead of Frechet Regression | Conditional prediction · NYC taxi data | Affine-invariant: 8.0099 -> 0.6123 | Helped |
| Use conditional SPD-DDPM instead of DDPM | Conditional prediction · NYC taxi data | Frobenius: 62.97 -> 6.4684 | Helped |
| Increase $\gamma$ from 2 to 4 (ablation) | SPD-DDPM · Toy data, unconditional generation | Mean distance: 65.81$\pm$ 17.86 -> 35.99$\pm$ 7.58 | Helped |
| Increase $\gamma$ from 4 to 10 (ablation) | SPD-DDPM · Toy data, unconditional generation | Mean distance: 35.99$\pm$ 7.58 -> 23.17$\pm$ 3.00 | Helped |
| Add double convolution (ablation) | SPD-DDPM with SPD U-Net · Toy data, unconditional generation | Mean distance: 26.17$\pm$ 3.75 -> 23.17$\pm$ 3.00 | Helped |
| Add concatenation structure (ablation) | SPD-DDPM with SPD U-Net · Toy data, unconditional generation | generation success | Helped |
| Add down and up convolutions (ablation) | SPD-DDPM with SPD U-Net · Toy data, unconditional generation | Mean distance: 42.75$\pm$ 3.25 -> 23.17$\pm$ 3.00 | Helped |
Surprising findings
- Standard DDPM performs very poorly on SPD data (Frobenius 62.97, affine-invariant 1203.10 on taxi data), indicating Euclidean diffusion does not transfer to SPD space.
- Removing the concatenation structure makes generation fail, per the authors.
Limitations
- Evaluation limited to toy synthetic data and a single taxi dataset with a small number of samples.
- Frobenius metric reported in Tab. for conditional setting is ambiguously defined (sqrt of sum of squares), and the Frechet Regression comparison uses a different optimized metric.
- Theoretical derivations rely on informal manipulations; bound on KL is stated with an unspecified constant.
Experimental setup
Datasets: synthetic SPD matrices sampled from Gaussian in SPD space NYC TLC taxi trip records 2019 with weather and trip predictors
Benchmarks: synthetic SPD toy data NYC taxi data
Metrics: mean affine-invariant distance Frobenius distance affine-invariant distance
Baselines: DDPM Frechet Regression
Builds on: DDPM SPDNet Fréchet Regression classifier-free guidance
Closest papers by meaning
- UDPM: Upsampling Diffusion Probabilistic Models (NeurIPS 2024 · similarity 0.89)
- O(d/T) Convergence Theory for Diffusion Probabilistic Models under Minimal Assumptions (ICLR 2025 · similarity 0.89)
- Denoising Diffusion Step-aware Models (ICLR 2024 · similarity 0.88)
- Assessing the quality of denoising diffusion models in Wasserstein distance: noisy score and optimal bounds (NeurIPS 2025 · similarity 0.88)
- Heavy-Tailed Diffusion with Denoising Levy Probabilistic Models (ICLR 2025 · similarity 0.87)
- Diffusion Models are Secretly Exchangeable: Parallelizing DDPMs via Auto Speculation (ICML 2025 · similarity 0.87)
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