Efficient Diffusion Models for Symmetric Manifolds

Authors: Oren Mangoubi, Nisheeth K. Vishnoi
Primary area: probabilistic methods (Bayesian methods, variational inference, sampling, UQ, etc.)
Keywords: Diffusion model, symmetric manifolds

Abstract:
We present a  framework for designing  efficient diffusion models on symmetric Riemannian manifolds, which include the  torus, sphere, special orthogonal group, and unitary group. While diffusion models on symmetric manifolds have gained significant attention, existing approaches often rely on the manifolds' heat kernels, which lack closed-form expressions and result in exponential-in-dimension per-iteration runtimes during training. We introduce a new diffusion model for symmetric-space manifolds, leveraging a projection of Euclidean Brownian motion to bypass explicit heat kernel computations. Our training algorithm minimizes a novel objective function derived via Ito's Lemma, with efficiently computable gradients, allowing each iteration to run in polynomial time for symmetric manifolds. Additionally, the symmetries of the manifold ensure the diffusion satisfies an "average-case" Lipschitz condition, enabling accurate and efficient sample generation. These improvements enhance both the training runtime and sample accuracy for key cases of symmetric manifolds, helping to bridge the gap between diffusion models on symmetric manifolds and Euclidean space.

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