arXiv:2609.30727v2 Announce Type: replace Abstract: Iterative algorithms for inverse imaging problems split reconstruction into alternating subproblems, one of which requires evaluating the data-fitting proximal map. In many practical cases, this proximal map has no analytic solution and so must be approximated by an inner loop solver at every iteration, compounding the cost of the outer reconstruction loop. To address this, we propose the pseudo-proximal map (PPM), a reformulation of the data-fitting proximal map as the minimum mean square error estimate of a synthetic probabilistic model. We implement the deep PPM as a self-supervised neural network trained only on sampled Gaussian noise, requiring no ground-truth training images. The deep PPM can be trained for any operator for which the forward model $A$ and its transpose $A^T$ can be evaluated, with provable equivalence to the proximal map when $A$ is linear. We validate the deep PPM on Gaussian deblurring and 4x super-resolution, where the proximal map has an analytic solution, and on X-ray computed tomography (XCT), where no analytic solution exists. Used as the data-fitting agent within an iterative reconstruction method, the deep PPM reproduces the reference reconstruction to within 1% NRMSE for all three operators, and for XCT, it replaces the inner conjugate-gradient loop with a single network evaluation that is 18x faster.