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.

翻译：生成扩散模型以及科学与工程中的许多随机模型在离散化前天然存在于无限维空间中。为将观测数据融入统计与学习任务，需对过程进行条件化处理。尽管近期研究已处理无限维线性过程的条件化问题，但无限维非线性过程的条件化尚未得到探索。本文直接对函数值随机过程进行条件化处理，无需预先离散化。为此，我们采用无限维版本的Girsanov定理对函数值随机过程施加条件约束，推导出包含得分函数的条件过程随机微分方程（SDE）。我们将此技术应用于进化生物学中生物体形状的时间序列分析，通过傅里叶基进行离散化，并利用得分匹配方法学习得分函数的系数。