CauchyNet: Compact and Data-Efficient Learning using Holomorphic Activation Functions

Abstract

CauchyNet is a compact complex-valued network whose hidden units form products of shifted reciprocal features, motivated by Cauchy-type kernel representations. The design targets regression problems with sharp rational-like spikes and partially observed inputs, where standard real-valued networks often require large width. We prove that finite linear combinations of multivariate Cauchy kernels are dense in $C(M)$ on compact $M\subset\mathbb R^N$, and that CauchyNet can realize these kernel sums via its complex biases and output weights. In experiments, CauchyNet reaches lower error on near-singular and gap-filling benchmarks, often with substantially fewer trainable parameters in our settings; results on smooth and piecewise-affine targets are mixed and delineate the method's intended regime.