发表机构
University of Toronto; University of Washington; Courant Institute School of Mathematics, Computing, and Data Science（多伦多大学; 华盛顿大学; 库朗数学科学研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出一类通过有限维映射与参考测度关联的函数空间测度，利用块三角传输映射实现低维采样，并给出逼近理论与误差分解，应用于逆问题和生成建模。
AI 中文摘要
函数空间上的测度广泛出现在贝叶斯逆问题和生成建模中，通常相对于一个可处理的参考测度具有低维结构。我们引入了测度类 $\mathcal{P}_\psi(\mu)$，该类中的测度仅通过一个有限维映射 $\psi$ 与参考测度 $\mu$ 不同，同时保留参考测度在其纤维上的条件分布。类成员由其 $d$ 维推前测度（在 $\psi$ 下）决定，并具有方便的块三角传输映射表示。样本从该 $d$ 维分布中抽取，然后通过采样参考条件分布补全到函数空间。对于高斯参考，这些传输映射是恒等映射的有限秩扰动。相比之下，最优传输映射不保留这种低维结构。对于在参考的Cameron-Martin几何中为迹类的协方差扰动，我们证明了最优映射是恒等映射的迹类扰动。即使有限秩扰动产生的修正，其秩由不变子空间决定，通常是无限的。我们发展了 $\mathcal{P}_\psi(\mu)$ 的逼近理论以及在其内部拟合的误差分析，将总误差分解为不可约的类误差和由 $\psi$ 的维数（而非环境离散化）决定的有限维边际项。我们给出了数值结果，包括从低维非线性观测映射进行推断、跳跃过程先验下的反卷积，以及Navier-Stokes流的状态估计。
英文摘要
Measures on function spaces arise throughout Bayesian inverse problems and generative modeling, often with low-dimensional structure relative to a tractable reference measure. We introduce the class $\mathcal{P}_ψ(μ)$ of measures that differ from a reference measure $μ$ only through a finite-dimensional map $ψ$ while preserving the reference conditionals on its fibers. Class members are determined by their $d$-dimensional pushforwards under $ψ$ and admit convenient block-triangular transport map representations. Draws are taken from this $d$-dimensional distribution and then completed to function space through sampling of the reference conditionals. For Gaussian references, these transport maps are finite rank perturbations of the identity. In contrast, optimal transport maps do not preserve this low-dimensional structure. For covariance perturbations that are trace class in the Cameron-Martin geometry of the reference, we show the optimal map is a trace class perturbation of the identity. Even finite rank perturbations yield corrections whose rank, governed by an invariant subspace, is typically infinite. We develop approximation theory for $\mathcal{P}_ψ(μ)$ and error analysis for fitting within it, splitting the total error into an irreducible class error and a finite-dimensional marginal term set by the dimension of $ψ$ rather than the ambient discretization. We present numerical results including inference from low-dimensional nonlinear observation maps, deconvolution under a jump process prior, and state estimation for Navier-Stokes flows.
