mlsanityData to 14 Oct 2026

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

arXiv 2606.04390 ↗First posted 3 Jun 2026cs.LG cond-mat.dis-nn math.PRcited by 0

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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.

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References · 14

  1. 2020Fourier Neural Operator for Parametric Partial Differential Equationscited by 2,908
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  3. 2014Identifying and attacking the saddle point problem in high-dimensional non-convex optimizationcited by 289
  4. 2024Bayes-optimal learning of an extensive-width neural network from quadratically many samplescited by 23
  5. 2025Diffusion Models and the Manifold Hypothesis: Log-Domain Smoothing is Geometry Adaptivecited by 20
  6. 2023Typical and atypical solutions in non-convex neural networks with discrete and continuous weightscited by 13
  7. 2020Complex-Valued vs. Real-Valued Neural Networks for Classification Perspectives: An Example on Non-Circular Datacited by 12
  8. 2018Evaluation of Complex-Valued Neural Networks on Real-Valued Classification Taskscited by 9
  9. 2022Complex networks with complex weightscited by 9
  10. 2024Spectral Informed Neural Network: An Efficient and Low-Memory PINNcited by 6
  11. 2024Fast Ergodic Search with Kernel Functionscited by 5
  12. 2023High-dimensional manifold of solutions in neural networks: insights from statistical physicscited by 4
2 more
  1. 2023Complex Network for Complex Problems: A comparative study of CNN and Complex-valued CNNcited by 2
  2. 2025Storage capacity of perceptron with variable selectioncited by 1

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