I am currently studying the solutions to time-reversed noising diffusion processes. These are the kind that are driven by the noised score of the target distribution we aim to sample from. In particular, when the data distribution is supported on a lower dimensional embedded submanifold (which is the case for the Manifold Hypothesis), there is some awkwardness when it comes to actually reaching the target distribution, as is it singular with respect to the ambient Euclidean Lebesgue measure. Things get even more complicated when the manifold is not known, which we call the implicit manifold case. Unfortunately for practitioners, this is often the case in machine learning. Luckily for me, it makes for an interesting problem.
Therefore, I am particularly keen on both constructing diffusion processes on implicit manifolds (see the "Implicit Manifold Diffusions" tab below), as well as understanding the behaviour of generative diffusions as an embedded geometry problem. The former mostly consists of limiting Dirichlet forms, and their connection to diffusion Markov processes. The latter mostly consists of stochastic differential geometry. Work on the latter is in progress.
This paper covers how to construct diffusion processes on data manifolds. See the full blog post here.
Beyond the geometric perspective, I am also interested in sampling from dynamical systems (Euclidean and manifold-valued), with some first work available here.
I also have a very infrequent YouTube channel, where i talk about whatever i feel like. I've made a few albums e.g. here, some of which you can find on my Spotify.