Shortcomings and capacities of real-constrained neural networks in complex spaces
Title: Evaluating the Limitations and Capabilities of Real-Constrained Neural Networks in Complex Domains
Abstract: This study determines the asymptotic ratio of storage capacities when real pre-activations are imposed within a complex hypothesis class, comparing this scenario to one where the entire class remains complex. Our analytical approach is grounded in Gardner volume comparisons performed at the threshold of critical capacity. Uniquely, our proof incorporates the Harish-Chandra-Itzykson-Zuber (HCIZ) formula, a technique that is unconventional in existing literature. By leveraging the HCIZ formula, we achieve a more resilient approximation of the final asymptotic ratio. This methodology is particularly well-suited to our research, as it enables integration over unitary and orthogonal compact manifolds, a process streamlined through the use of the Weyl integration formula and the Haar measure.
Source: arXiv Generated at: 2026-06-04 00:00:00 UTC
