Summary
The theory of stochastic differential equations on vector spaces forms the basis for recent diffusion generative models, and recent efforts have attempted to extend this framework to stochastic differential equations on Riemannian manifolds (Riemannian diffusion models). From a practical computational perspective this extension is less than straightforward as pointed out in this paper, e.g., computing things like "straight lines" (minimal geodesics) become very difficult two-point boundary value problems; quantities that admitted closed-form analytic formulas in the vector space setting do not translate to closed-form formulas in the Riemannian setting. Past attempts at practical work-arounds and computational approximations are limited to low-dimensional cases; these methods do not scale well to higher-dimensional spaces. By restricting the focus to Riemannian manifolds that are also symmetric spaces, the authors present a collection of computation and approximation techniques that, taken together, demonstrate both computational tractability and scalability. Experiments validating these claims are presented.
Strengths
The paper is technically sound and well-written, and the contributions are laid out in a clear and concise manner without claiming too much. Rather than making claims about the significance of a particular result, I appreciate that the authors claim that it is the sum of a set of smaller contributions that, taken together, lead to a computationally practical, scalable set of techniques to make Riemannian diffusion models workable in practice. The focus on Riemannian symmetric spaces seems reasonable, and the detailed computation formulas and approximations are laid out in enough detail that they can be implemented by a reader familiar with stochastic differential equations and some Riemannian geometry. The examples and experiments appear to support the claims of the authors about computational tractability and scalability to higher-dimensional problems.
Weaknesses
-The following is a general comment applicable not only to this paper, but to all papers that present a “geometric” version of an existing algorithm or method in machine learning. In the past, developing a geometric, manifold version of a vector space algorithm could be regarded as a meaningful contribution in itself, and I would argue that the Riemannian diffusion model is somewhat in this spirit (equivariant models I would argue are not in this spirit however). Of course, it’s hard to argue against the claim that the Riemannian diffusion model is needed for problems in which the underlying data are manifold-valued; in that case I would be much more convinced by examples and case studies drawn from mainstream applications rather than narrow ones. As geometric methods have become more mainstream, the threshold for what constitutes a meaningful contribution is justifiably higher: given the much more difficult computations involved in computing, e.g., derivatives, gradients, Laplacians, minimal geodesics, kernels, etc., the considerable increase in computation needs to be justified by results. For this paper, the authors recognize and point out the difficulty of computing the geometric quantities and propose more efficient approximations, which is worthwhile (but whether they deserve to be published in NeurIPS is another matter). It would be helpful if the authors can address this question more directly – are the added computational difficulties justified by a commensurate improvement in the results? The experiments do not strike me as mainstream problems, for one thing. (Note: This comment could also have been placed in the "questions" section, but I place it here as it could be a potential weakness).
-The definition of symmetric space could be sharpened, as this is an important underlying assumption in the paper, e.g., clarify the isometry requirement (isometry between what spaces?), make the distinction between local vs global symmetric space, provide brief intuition (reflexive symmetry about a point), and most importantly, provide examples of spaces arising in ML applications that are symmetric (the authors list some -- a few more would be helpful, e.g., the space of symmetric positive-definite matrices, which is nonimpact but a space that arises constantly in ML) and also examples of non-symmetric spaces (e.g., hyperbolic manifolds).
-The assumption of the Riemannian manifold being embedded in a higher-dimensional ambient Euclidean space in itself is not restrictive thanks to Whitney and Nash, and most practical problems that I've encountered usually admit some natural embedding. The compactness assumption, however, could be somewhat restrictive for certain problems in which the underlying data manifold is potentially unbounded. The authors brush aside this case by asserting that noncompact Riemannian manifolds are diffeomorphic to Euclidean space, and therefore the standard way of using the embedding in R^n to model diffusion, then mapping back to the surface, is sufficient. One could then ask why this approach doesn't work for any manifold embedded in Euclidean space; it would be helpful if the authors could clarify this point.
Questions
-Explaining in more detail the Riemannian exponential map would be helpful, so that the reader has a better idea of what the computation entails.
-The examples are for SU(3) and the hypersphere, which are rather specific manifolds. Particularly in the case of the hypersphere, I suspect that the heavy Riemannian diffusion machinery may not be needed to arrive at the result, since Brownian motion on the sphere is well-characterized and projections to the hypersphere are trivial.
Rating
6: Weak Accept: Technically solid, moderate-to-high impact paper, with no major concerns with respect to evaluation, resources, reproducibility, ethical considerations.
Confidence
4: You are confident in your assessment, but not absolutely certain. It is unlikely, but not impossible, that you did not understand some parts of the submission or that you are unfamiliar with some pieces of related work.
Limitations
No potential negative societal impact beyond that of other typical NeurIPS submissions that I can detect.