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Efficient Diffusion Models for Symmetric Manifolds
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Proposes an efficient diffusion model for symmetric-space manifolds (torus, sphere, SO(n), U(n)) using a projected Ornstein-Uhlenbeck forward process with spatially varying covariance and an Ito-derived training objective, with per-iteration cost O(1) gradient evaluations and near-linear arithmetic cost.
Problem
Diffusion models on symmetric-space Riemannian manifolds rely on heat kernels lacking closed forms, so training needs exponential-in-d arithmetic or O(d) gradient evaluations per step, and sampling guarantees are not polynomial in dimension.
Approach
Define the forward diffusion as the projection of a Euclidean Ornstein-Uhlenbeck process onto the manifold, which gives a spatially varying covariance; derive the reverse SDE on the manifold via Ito's lemma so the training objective uses only the Euclidean Gaussian kernel, and exploit symmetry so the covariance is parameterized by n^2 scalars.
Key results
- On the torus, the model's NLL is lowest for d>=10 (e.g., 0.97 vs 2.51 for RSGM at d=1000).
- On U(n) with n>=9 and SO(n) with n>=9, the model achieves the lowest C2ST scores (e.g., SO(n) n=15: 0.67 vs 0.72 Euclidean and 0.79 TDM).
- Per-iteration training runtime on U(n) at n=50 is 0.60 s vs 11.55 s for RSGM and 9.43 s for TDM, staying within ~3x of the Euclidean model.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use the proposed manifold diffusion instead of RSGM | Torus wrapped Gaussian · Torus, d=10, 50, 100, 1000 | NLL: 0.49\!\pm\!.05 -> 0.47\!\pm\!.03; 0.62\!\pm\!.06 -> 0.50\!\pm\!.05; 1.45\!\pm\!.21 -> 0.52\!\pm\!.09; 2.51\!\pm\!.17 -> | Helped |
| Use the proposed manifold diffusion instead of RSGM | Torus wrapped Gaussian · Torus, d=2 | NLL: 0.42\!\pm\!.03 -> 0.49\!\pm\!.04 | Hurt |
| Use the proposed manifold diffusion instead of the Euclidean diffusion model | U(n) quantum oscillator · U(n), n=3, 5, 10, 30, 50 | per-iteration runtime (seconds): 0.19\pm\!.01 -> 0.36\pm\!.00; 0.19\pm\!.01 -> 0.36\pm\!.00; 0.20 \pm\!.01 -> 0.36\pm\!.00; 0.20\pm\!.01 -> 0.46\pm\!.01; | Hurt |
| Use the proposed manifold diffusion instead of RSGM | U(n) quantum oscillator · U(n), n=50 | per-iteration runtime (seconds): 11.55\pm\!.31 -> 0.60\pm\!.01 | Helped |
| Use the proposed manifold diffusion instead of TDM | U(n) quantum oscillator · U(n), n=50 | per-iteration runtime (seconds): 9.43\pm\!.23 -> 0.60\pm\!.01 | Helped |
| Use the proposed manifold diffusion instead of the Euclidean diffusion model | SO(n) wrapped Gaussian mixture · SO(n), n=3, 5 | C2ST: .51.\pm\!.01 -> .55.\pm\!.01; .51.\pm\!.01 -> .56.\pm\!.02 | Hurt |
| Use the proposed manifold diffusion instead of the Euclidean diffusion model | SO(n) wrapped Gaussian mixture · SO(n), n=9, 12, 15 | C2ST: .62 .\pm\!.02 -> .60.\pm\!.03; .64.\pm\!.02 -> .61.\pm\!.02; .72.\pm\!.02 -> .67.\pm\!.03 | Helped |
| Use the proposed model instead of prior manifold diffusion models | Symmetric-manifold diffusion training · U(n) and SO(n), n=50 | per-iteration runtime: 45 and 57 times greater; 51 and 66 times greater | Helped |
Surprising findings
- Euclidean diffusion with projection performs worse than manifold-native methods on torus at high dimension (NLL 8.7 vs 0.97 at d=1000).
Limitations
- Sampling guarantees are poly(d) only for the stated symmetric manifolds under an average-case Lipschitz assumption; general manifolds not covered.
- Experiments are proof-of-concept on synthetic datasets only; log-likelihood not computed for SO(n)/U(n) at larger n.
- SO(n) analysis requires additional handling of exponentially small eigengaps.
Experimental setup
Datasets: wrapped Gaussian on torus (synthetic) wrapped Gaussian mixture on SO(n) (synthetic) quantum oscillator time-evolution operators on U(n) (synthetic)
Benchmarks: wrapped Gaussian torus NLL U(n) quantum evolution C2ST SO(n) wrapped Gaussian mixture C2ST
Metrics: negative log-likelihood C2ST per-iteration training runtime (s)
Baselines: RSGM TDM Euclidean diffusion model RDM SCRD
Builds on: Riemannian score-based generative models Trivialized momentum diffusion denoising diffusion models
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