Machine learning in non-Euclidean spaces has been rapidly attracting attention in recent years, and this talk will give some examples of progress on its mathematical and algorithmic foundations. A sequence of developments that eventually leads to non-Euclidean generative modeling will be reported.
More precisely, I will begin with variational optimization, which, together with delicate interplays between continuous- and discrete-time dynamics, enables the construction of momentum-accelerated algorithms that optimize functions defined on manifolds. Selected applications, namely a generic improvement of Transformer, and a low-dim. approximation of high-dim. optimal transport distance, will be described. Then I will turn the optimization dynamics into an algorithm that samples from probability distributions on Lie groups. This sampler probably converges, even without log-concavity condition or its common relaxations. Finally, I will describe how this sampler can lead to a structurally-pleasant diffusion generative model that allows users to, given training data that follow any latent statistical distribution on a Lie group, generate more data exactly on the same manifold that follow the same distribution. If time permits, applications such as molecule design and generative innovation of quantum processes will be briefly discussed.