Approximating Measures on Function Spaces: Transport and Truncation
TL;DR
This work introduces the class $\mathcal{P}_\psi(\mu)$ of measures that differ from a reference measure only through a finite-dimensional map $\psi$ while preserving the reference conditionals on its fibers, and develops approximation theory for fitting within it.
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 $\mathcal{P}_\psi(\mu)$ of measures that differ from a reference measure $\mu$ only through a finite-dimensional map $\psi$ while preserving the reference conditionals on its fibers. Class members are determined by their $d$-dimensional pushforwards under $\psi$ 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 $\mathcal{P}_\psi(\mu)$ 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 $\psi$ 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.