Stochastic Numerics for Sampling on Manifolds

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The digital era has led to the increasing availability of highly-structured data such as social media graphs and networks, ratings and recommender system data from online retail and streaming platforms, and high-resolution medical images. Such data are characterised by non-trivial constraints (not everyone but only friends and family form a group within a network; shape of the imaged brain is unchanged under rotations of the image), and the sheer scale and complexity associated with storing and analysing such data necessitate the use of probabilistic models to mimic the manner in which the data were generated. Fundamental to successful practical use of probabilistic models for highly-structured data is sampling, or generating random data, from geometrically constrained spaces known as manifolds. State-of-the-art in efficient sampling, backed by theoretical guarantees, within this nascent area is restricted to cases where the manifold is smooth without a boundary or the sampling distribution belongs to a class that is particularly amenable for theoretical analysis. This excludes many important problems one routinely encounters in AI and statistical applications, including low-rank matrix completion (predicting user ratings for Netflix movies) and analysing shapes of objects (computing a representative tumour shape from medical images). To this end, the overarching goal of this timely project is to develop and analyse methods to sample from a general class of manifolds and distributions using ergodic stochastic differential equations. Positioned at the interface of stochastics, numerical analysis and geometry, the project will make a major contribution to the advancement of numerical methods for SDEs on manifolds and thus open up the possibility to efficiently analyse complex, geometric data.