  Generative models based on flow matching have attracted significant attention
for their simplicity and superior performance in high-resolution image
synthesis. By leveraging the instantaneous change-of-variables formula, one can
directly compute image likelihoods from a learned flow, making them enticing
candidates as priors for downstream tasks such as inverse problems. In
particular, a natural approach would be to incorporate such image probabilities
in a maximum-a-posteriori (MAP) estimation problem. A major obstacle, however,
lies in the slow computation of the log-likelihood, as it requires
backpropagating through an ODE solver, which can be prohibitively slow for
high-dimensional problems. In this work, we propose an iterative algorithm to
approximate the MAP estimator efficiently to solve a variety of linear inverse
problems. Our algorithm is mathematically justified by the observation that the
MAP objective can be approximated by a sum of $N$ ``local MAP'' objectives,
where $N$ is the number of function evaluations. By leveraging Tweedie's
formula, we show that we can perform gradient steps to sequentially optimize
these objectives. We validate our approach for various linear inverse problems,
such as super-resolution, deblurring, inpainting, and compressed sensing, and
demonstrate that we can outperform other methods based on flow matching.
