Data-driven sampling and prediction on manifolds
Résumé
With the possibility of interpreting data using increasingly complex models, we are often faced with the need to embed the data in an ambient space consistent with the parameterization of these models, typically a high-dimensional Euclidean space. Constructing probability measures on these spaces or subsets of them is fairly straightforward once the subsets have been delineated. Quite often, fundamental laws, associated for example with symmetry of conservation, constrain the data to a complex manifold within this ambient space. Acknowledgning these constraints serves to focus the scatter in the data around the manifold with significant ramifications to subsequent statistical analysis: the shape of the distributions, asymptotic sample properties, and the sampling mechanisms would all be affected. Increasingly more often, the exact constraints (hence manifolds) to which the data is subjected are not known, either because of unaccounted interaction with other scales or physics (such as in physics-based problems), or because the fundamental governing laws are not yet understood (such as in biological, social, and economical systems). In such situations, manifold discovery is an important step in augmenting statistical analysis with key hidden constraints. In this talk, we present a recent procedure for describing probability measures on diffusion manifolds and sampling from them. The procedure integrates methods from machine learning with statistical estimation, functional analysis, and white noise calculus to achieve orders of magnitude efficiencies in data requirements for probabilistic characterization and sampling. Examples will be shown from applications from across the sciences and engineering.