LOCUS: Feasible-Centered Null-Space Flow for Inverse Problems

Inverse problems combine incomplete measurements with prior knowledge, from structural assumptions in compressed sensing to learned generative priors. For noiseless linear measurements $y=Ax$, all measurement-consistent reconstructions lie in the affine fiber $A^{†}y+\ker (A)$. Existing measurement-consistent constructions use this geometry to determine where generative transport may move, and raise a distinct design question: where within this feasible fiber should that transport be centered? We propose to predict an observation-dependent reconstruction, project it onto the feasible fiber, and use flow matching to generate the remaining null-space residual. This separates three roles: preserving what is measured, predicting what is inferable, and generating what remains after that prediction. The resulting dynamics preserve measurements throughout sampling. For independent projected-Gaussian sources and linear training paths, we show that excess center error relative to the conditional mean exactly equals excess displacement energy and ideal marginal-flow action. This gives prediction quality a direct interpretation in terms of residual transport. Across 17 large-cohort evaluations on FFHQ, CelebA-HQ, and ImageNet, spanning block and random-pixel inpainting and $4\times$–$16\times$ super-resolution, the proposed design improves both PSNR and LPIPS over the corresponding pixel-centered null-space flow. A controlled 512-image study separates the center's conditioning and source roles, while a separate 2,000-image random-pixel benchmark demonstrates strong reconstruction and perceptual quality at 20 flow evaluations.