Discontinuous Prior-Mode Sections and the Geometry of Ambiguity in Intrinsic Image Decomposition

What the math gives to ML

The paper models ambiguity as a discontinuous prior-mode section: two qualitatively different decompositions are preferred on opposite sides of an ambiguity boundary. Its transferable asset is an explicit scaling law for how smooth neural models represent that discontinuity, namely a transition-layer Jacobian or latent-curve curvature proportional to the branch jump divided by the square root of the smoothness strength. This suggests both a practical ambiguity diagnostic and an architectural intervention: detect such singular regions and replace a single smoothly interpolating predictor with explicit branches plus a learned gate. The first useful test is on intrinsic-image, inverse-rendering, or any multimodal inverse problem where smooth averaging is known to produce visibly wrong outputs.