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

arXiv: https://arxiv.org/abs/2606.04390 · HTML: https://mlsanity.com/paper/2606.04390/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 14 of its 70 references that are in the map.
- First posted on arXiv 3 Jun 2026.

## Builds on (most cited first, 14 of 14)

- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Beyond neural scaling laws: beating power law scaling via data pruning](https://mlsanity.com/paper/2206.14486/) (2022)
- [Identifying and attacking the saddle point problem in high-dimensional non-convex optimization](https://mlsanity.com/paper/1406.2572/) (2014)
- [Bayes-optimal learning of an extensive-width neural network from quadratically many samples](https://mlsanity.com/paper/2408.03733/) (2024)
- [Diffusion Models and the Manifold Hypothesis: Log-Domain Smoothing is Geometry Adaptive](https://mlsanity.com/paper/2510.02305/) (2025)
- [Typical and atypical solutions in non-convex neural networks with discrete and continuous weights](https://mlsanity.com/paper/2304.13871/) (2023)
- [Complex-Valued vs. Real-Valued Neural Networks for Classification Perspectives: An Example on Non-Circular Data](https://mlsanity.com/paper/2009.08340/) (2020)
- [Evaluation of Complex-Valued Neural Networks on Real-Valued Classification Tasks](https://mlsanity.com/paper/1811.12351/) (2018)
- [Complex networks with complex weights](https://mlsanity.com/paper/2212.06257/) (2022)
- [Spectral Informed Neural Network: An Efficient and Low-Memory PINN](https://mlsanity.com/paper/2408.16414/) (2024)
- [Fast Ergodic Search with Kernel Functions](https://mlsanity.com/paper/2403.01536/) (2024)
- [High-dimensional manifold of solutions in neural networks: insights from statistical physics](https://mlsanity.com/paper/2309.09240/) (2023)
- [Complex Network for Complex Problems: A comparative study of CNN and Complex-valued CNN](https://mlsanity.com/paper/2302.04584/) (2023)
- [Storage capacity of perceptron with variable selection](https://mlsanity.com/paper/2512.01861/) (2025)

## Abstract (arXiv)

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.