Inverse Problem Solving by Walking the Sphere

Latent-space inverse problem solving searches a generator's latent space for a code whose output matches the measurement, making data fit the only objective. With GAN, VAE, and normalizing-flow priors, this recovers signals well, but the set of codes the generator decodes reliably is known only implicitly, so an iterate that drifts away from it can only be pulled back by a penalty that competes with the data fit. Sphere encoders make this set explicit: their latents lie on a hypersphere by construction, and a single normalization returns any iterate to it. We solve inverse problems by projected gradient descent on this sphere, a method we call Walk the Sphere (WTS), and introduce a variant, WTS+Loop, that periodically re-encodes the iterate. Against five diffusion-based solvers tuned under the same budget, WTS or WTS+Loop ranks first, outright or tied, in 13 of 18 settings spanning ImageNet-256 and FFHQ-256 restoration and three scientific problems from InverseBench. The solver needs only three hyperparameters and 300 gradient steps, taking about two seconds per image on FFHQ-256.