Conditioning non-linear and infinite-dimensional diffusion processes
Elizabeth Louise Baker Gefan Yang Michael L. Severinsen Christy Anna Hipsley Stefan Sommer
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.