Conditioning non-linear and infinite-dimensional diffusion processes

TL;DR

This paper develops a method to condition non-linear, infinite-dimensional stochastic processes directly, enabling advanced modeling of complex functional data without prior discretization, with applications in evolutionary biology.

Contribution

It introduces an infinite-dimensional Girsanov's theorem-based approach for conditioning non-linear processes, extending existing linear process methods to more complex, function-valued stochastic models.

Findings

Successfully conditioned non-linear infinite-dimensional processes

Applied method to shape analysis in evolutionary biology

Discretized processes via Fourier basis and learned scores

Abstract

Generative diffusion models and many stochastic models in science and engineering naturally live in infinite dimensions before discretisation. To incorporate observed data for statistical and learning tasks, one needs to condition on observations. While recent work has treated conditioning linear processes in infinite dimensions, conditioning non-linear processes in infinite dimensions has not been explored. This paper conditions function valued stochastic processes without prior discretisation. To do so, we use an infinite-dimensional version of Girsanov's theorem to condition a function-valued stochastic process, leading to a stochastic differential equation (SDE) for the conditioned process involving the score. We apply this technique to do time series analysis for shapes of organisms in evolutionary biology, where we discretise via the Fourier basis and then learn the coefficients of the score function with score matching methods.