Shortcomings and capacities of real-constrained neural networks in complex spaces

ArXi:2606.04390v1 Announce Type: new We find the asymptotic ratio between the storage capacities when enforcing real pre-activations in a complex hypothesis class as opposed to complex ones in the same class. Our methods depend on Gardner volume comparisons at critical capacity. Our proof relies on an application of the Harish-Chandra-Itzykson-Zuber (HCIZ) formula, nonstandard in literature. With the HCIZ formula, we may obtain a robust approximation for the final asymptotic ratio.