Cauchy-Riemann Regularization for Extrapolation and Structured Representations

Out-of-support extrapolation and controllable representation learning are usually treated as separate problems, yet both require models to capture the constraints underlying the training distribution rather than merely fit its observed support. Existing methods rely on smoothness, sparsity, orthogonality, or distribution-specific penalties, but the connection between these penalties and the geometry of the data distribution and learned function class is often indirect. We ask whether classical complex analysis can provide design principles for distribution-agnostic geometric regularization. For maps $f:\mathbb{C}^{k}\to\mathbb{C}^{m}$, the Cauchy–Riemann equations are equivalent to the vanishing of the Wirtinger conjugate derivative, $\partial_{\bar{z}} f = 0$, a condition with two leveragable consequences: (1) in the holomorphic regime, analytic continuation constrains predictions outside the training support, and (2) locally, complex-linearity couples paired real–imaginary directions through orthogonality and matched scaling. We turn this condition into a differentiable regularizer for split-real neural networks, so that a single regularizer can both direct predictable continuation and impose structure on learned representations. In particular, it promotes conformality by preserving angles and local scales, and it supports equivariance to transformations represented by complex multiplication.