arXivDaily arXiv每日学术速递 周一至周五更新
arXiv 2402.01434stat.MLcs.LGstat.CO

非线性与无限维扩散过程的条件化

Conditioning non-linear and infinite-dimensional diffusion processes

University of Copenhagen(哥本哈根大学)

Elizabeth Louise Baker, Gefan Yang, Michael L. Severinsen, Christy Anna Hipsley, Stefan Sommer

AI总结:

本文提出利用无限维Girsanov定理对函数值随机过程进行条件化,得到涉及得分的SDE,并应用于进化生物学形状时间序列分析,通过傅里叶基离散化和得分匹配实现。

摘要:

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。为了在统计和学习任务中纳入观测数据,需要对观测进行条件化。尽管近期工作已处理无限维线性过程的条件化,但无限维非线性过程的条件化尚未被探索。本文在不进行先验离散化的情况下,对函数值随机过程进行条件化。为此,我们使用无限维版本的Girsanov定理来条件化函数值随机过程,从而得到涉及得分(score)的条件化过程的随机微分方程(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,其中我们通过傅里叶基进行离散化,然后使用得分匹配方法学习得分函数的系数。

英文摘要:

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.

机构、总结、中文摘要均由AI分析而来,可能存在错误,请以论文正文为准。

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