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

2026-06-02

AI Summary

This study investigates the impact of enforcing real-valued pre-activations on the storage capacity of complex-valued neural networks. By comparing the critical capacities derived from the Gardner volume under real constraints versus the fully complex case, the authors introduce—for the first time—the Harish-Chandra–Itzykson–Zuber integral formula in this context. Combined with Weyl’s integration formula and the Haar measure, this approach enables high-dimensional integration over unitary and orthogonal compact manifolds, establishing a more robust asymptotic analysis framework. The method precisely characterizes the asymptotic ratio of storage capacities between the two network types, quantifying the performance loss induced by real constraints and offering new theoretical tools for understanding complex-valued neural networks.

Abstract

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 more robust approximation for the final asymptotic ratio. This strategy is applicable to our work specifically since we integrate over the unitary and orthogonal compact manifolds, facilitated via the Weyl integration formula and the Haar measure.

Problem

• complex neural networks
• storage capacity
• real constraints
• pre-activations
• asymptotic ratio

Innovation

• HCIZ formula
• storage capacity
• complex neural networks
• Gardner volume
• Haar measure