Conditioning non-linear and infinite-dimensional diffusion processes
This work addresses a foundational gap in statistical and learning tasks for infinite-dimensional stochastic models, with potential applications in fields like evolutionary biology, though it appears incremental as it extends existing linear conditioning methods to non-linear cases.
The paper tackles the problem of conditioning non-linear, infinite-dimensional diffusion processes on observed data, which had not been explored before, and applies this to time series analysis of organism shapes in evolutionary biology by discretizing via the Fourier basis and learning score coefficients with score matching methods.
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.