Approximating Measures on Function Spaces: Transport and Truncation

2026-09-15

AI Summary

本文提出了一种通过有限维映射ψ来近似函数空间中测度的方法，该方法保留了参考测度的条件分布，并且适用于低维结构。

Abstract

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.

Problem

• Bayesian inverse problems
• generative modeling
• function spaces
• measures approximation
• low-dimensional structure

Innovation

• finite-dimensional map
• transport map
• Gaussian reference
• error decomposition
• function space measures