arXiv:2608.04540 2026 Regularization 1 ideas extracted · analyzed Aug 31, 2026
The paper provides a derivative-free local certificate for holomorphicity: a continuous complex field is holomorphic if its centered circular line integral is asymptotically smaller than the circle area, namely o(r^2) at every center. This can become a multiscale regularizer for complex-valued neural fields, enforcing Cauchy-Riemann structure without explicitly differentiating the network. The most credible use case is an analytic decoder or complex coordinate representation where holomorphicity is a desired inductive bias.
Add a multiscale circular-integral penalty to a complex-valued neural field f_theta: R^2 -> C. The penalty directly tests the local contour condition that characterizes holomorphic functions, providing a derivative-free alternative to explicitly penalizing the Cauchy-Riemann residual.