Shortcomings and capacities of real-constrained neural networks in complex spaces
This work provides theoretical insights into the capacity of neural networks with real-valued pre-activations in complex spaces, relevant for understanding constraints in complex-valued neural networks.
The paper derives the asymptotic ratio of storage capacities between real-constrained and unconstrained complex neural networks, using Gardner volume comparisons and the HCIZ formula to obtain a robust approximation.
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.