Anisotropic Higher-Order Semiregularity of Degenerate Generalized Equations

What the math gives to ML

The paper develops a graded notion of output perturbation for degenerate maps: target block Y_i is treated as reachable at scale t^i, rather than every direction being measured linearly. Its transferable asset is the anisotropic gauge q_p, together with a fixed-base inverse estimate that converts blockwise residual magnitudes into a source-space displacement bound. This suggests an inverse-network or implicit-layer training rule that normalizes residuals according to their local order of controllability instead of forcing all output coordinates into a Euclidean loss. The most direct test is on synthetic and small real inverse problems with rank-deficient Jacobians, where the method should improve stability and recovery of higher-order-controlled output directions.