Most of my researchstudies the generative/reverse dynamics of diffusion models. That is, a score-based SDE, and how its shape / solution changes across space and time.
The manifold hypothesis makes for an interesting case study, and I am currently trying to see whether or not we can analyse the phase transition induced by the dimensional collapse from ambient space to an embedded submanifold.
To do so, I mostly use tools from stochastic analysis, differential geometry, statistical mechanics & fields, as well as some occasional programming. You can find my first work treating this
in this paper, which constructs diffusion processes on manifolds only described by point clouds. More below.
This paper covers how to construct diffusion processes on data manifolds. See the full blog post here.
Implicit Manifold Diffusions
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.