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和工程中的许多随机模型在离散化之前自然存在于无限维空间中。为了将观测数据纳入统计和学习任务，需要对这些观测数据进行条件化。尽管最近的研究已经处理了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。本文在不进行预先离散化的情况下，对函数值随机过程进行条件化。为此，我们利用无限维版本的Girsanov定理对函数值随机过程进行条件化，从而得到一个包含得分（score）的条件化过程随机微分方程（SDE）。我们将该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。