Learning Piecewise-Smooth Image Structure with Implicit Regularization

In image regression, coefficient images are often piecewise smooth: approximately constant or smoothly varying within regions, with abrupt changes across boundaries. Explicit penalties such as total variation (TV) exploit this structure for spatially coherent, interpretable estimates but require a nonsmooth fit for every tuning value. To learn it through implicit regularization instead, we decompose the coefficient image into a polynomial background and a component reconstructed from local differences of a chosen order. With the background profiled out, a difference-of-squares factorization encourages sparse differences and a quadratic term keeps them consistent with an image. Gradient descent from a small balanced initialization, with backtracking and validation-based early stopping, fits the factors. Under correlated sub-Gaussian designs, we establish finite-step prediction and coefficient-error bounds. Under bounded variation, conservative steps yield an $n^{-1/2}$ prediction rate up to logarithmic and geometric factors, without restricted-eigenvalue assumptions. A validation oracle inequality supports checkpoint and order selection at logarithmic cost in the candidate count. The small-initialization gradient-flow limit connects the method to minimum-variation interpolation. Simulations and a sea surface temperature application show competitive accuracy at lower computational cost than TV in the studied settings.