Optimization-based samplers provide an efficient and parallellizable approach
to solving large-scale Bayesian inverse problems. These methods solve randomly
perturbed optimization problems to draw samples from an approximate posterior
distribution. "Correcting" these samples, either by Metropolization or
importance sampling, enables characterization of the original posterior
distribution. This paper presents a new geometric interpretation of the
randomize-then-optimize (RTO) method [1] and a unified transport-map
interpretation of RTO and other optimization-based samplers, i.e., implicit
sampling [19] and randomized-maximum-likelihood [20]. We then introduce a new
subspace acceleration strategy that makes the computational complexity of RTO
scale linearly with the parameter dimension. This subspace perspective suggests
a natural extension of RTO to a function space setting. We thus formalize a
function-space version of RTO and establish sufficient conditions for it to
produce a valid Metropolis-Hastings proposal, yielding dimension-independent
sampling performance. Numerical examples corroborate the dimension-independence
of RTO and demonstrate sampling performance that is also robust to small
observational noise.