Score learning and inference for diffusion processes on shape spaces

Steering diffusion processes towards a data distribution is an integral part of diffusion models in generative AI. For geometric data such as shape data, diffusion processes appear as models for stochastic dynamics of e.g. species change through evolution, or for generating data distributions on non-linear spaces, e.g. when defining constructs such as the diffusion mean that relies on geometric equivalents of the Gaussian distribution. Score learning is here key for conditioning on observed data. Thus, score learning provides a connection between generative models and geometric statistics. The talk will concern this connection, bridge simulation on geometric spaces, and application of score learning in geometric contexts. A specific example of this is conditioning diffusion processes in infinite dimensions allowing shape observations to be used for phylogenetic inference in evolutionary biology.