Shortcomings and capacities of real-constrained neural networks in complex spaces
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. We use weights drawn from the complex Gaussian, which converge asymptotically in norm to the square root of dimension almost surely. Our methods depend on Gardner volume-type 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.
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BibTeX
@misc{gracyk2026shortcomings,
title = {{Shortcomings and capacities of real-constrained neural networks in complex spaces}},
author = {Andrew Gracyk},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2606.04390},
eprint = {2606.04390},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.04390},
}
RIS
TY - UNPB
TI - Shortcomings and capacities of real-constrained neural networks in complex spaces
AU - Andrew Gracyk
PY - 2026
DA - 2026/09/02
PB - arXiv
DO - 10.48550/arxiv.2606.04390
UR - https://arxiv.org/abs/2606.04390
AB - 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. We use weights drawn from the complex Gaussian, which converge asymptotically in norm to the square root of dimension almost surely. Our methods depend on Gardner volume-type 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.
ER -