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arxiv: 2606.04390 · v1 · pith:6M2HJEXFnew · submitted 2026-06-03 · 💻 cs.LG · cond-mat.dis-nn· math.PR

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

Andrew Gracyk This is my paper

Pith reviewed 2026-06-28 06:50 UTC · model grok-4.3

classification 💻 cs.LG cond-mat.dis-nnmath.PR
keywords storage capacitycomplex neural networksreal pre-activationsGardner volumeHCIZ formulaasymptotic ratiohypothesis classcritical capacity
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The pith

Enforcing real pre-activations in a complex hypothesis class yields an asymptotic storage-capacity ratio relative to fully complex pre-activations.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper derives the asymptotic ratio of storage capacities between complex hypothesis classes restricted to real pre-activations and the same classes allowing complex pre-activations. It reaches this ratio by comparing Gardner volumes exactly at the critical capacity point. The derivation applies the Harish-Chandra-Itzykson-Zuber formula through integration over unitary and orthogonal manifolds via the Weyl formula and Haar measure. A reader would care because the ratio directly measures the capacity penalty imposed by the real constraint inside an otherwise complex model class.

Core claim

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 formula, nonstandard in literature, to obtain a more robust approximation for the final asymptotic ratio.

What carries the argument

Gardner volume comparisons at critical capacity, using the Harish-Chandra-Itzykson-Zuber formula integrated over unitary and orthogonal compact manifolds with the Weyl integration formula and Haar measure.

If this is right

  • The ratio supplies a precise quantitative limit on capacity loss from the real pre-activation constraint.
  • The same Gardner-volume plus HCIZ strategy applies to other activation constraints within complex hypothesis classes.
  • Fully complex pre-activations achieve strictly higher asymptotic storage capacity than real-constrained ones in the same class.
  • The approximation for the ratio is more robust than prior methods that did not invoke the HCIZ formula.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Designers of complex-valued networks could use the ratio to decide when the added cost of complex activations is justified by the capacity gain.
  • The result suggests testing whether other real-valued restrictions, such as real weights, produce similar capacity reductions.
  • Finite-size corrections to the asymptotic ratio might be derivable by extending the same manifold integration techniques.
  • The capacity gap implies that training algorithms for complex networks should prioritize preserving complex pre-activation statistics.

Load-bearing premise

The Harish-Chandra-Itzykson-Zuber formula applies validly when the integration is performed over unitary and orthogonal compact manifolds via the Weyl integration formula and Haar measure.

What would settle it

Numerical estimation of storage capacities for large finite networks with real versus complex pre-activations, checking whether their empirical ratio converges to the derived asymptotic value.

Figures

Figures reproduced from arXiv: 2606.04390 by Andrew Gracyk.

Figure 1
Figure 1. Figure 1: We plot Γ across varying ρ and G{r,c}, B. We suppress values of Γ ∈/ [0, 1] in the plot. In theory, G{r,c}, B are not free parameters, and Γ ∈/ [0, 1] is not an actual scenario. We take X to be plot value and B ′ = X − G′ r − ∂ρGc. We attempt to choose G ′ r , ∂ρGc, B ′ consistent with a real scenario. The void space is plausible mathematically but does not represent a realistic scenario in regard to physi… view at source ↗
Figure 2
Figure 2. Figure 2: We plot the same as Figure [PITH_FULL_IMAGE:figures/full_fig_p027_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: We plot the same as Figure [PITH_FULL_IMAGE:figures/full_fig_p027_3.png] view at source ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 1 minor

Summary. The manuscript claims to derive the asymptotic ratio between storage capacities of neural networks in a complex hypothesis class when enforcing real pre-activations versus allowing complex pre-activations. The derivation proceeds by comparing Gardner volumes at critical capacity and applies the Harish-Chandra-Itzykson-Zuber (HCIZ) formula, reduced via the Weyl integration formula and Haar measure over unitary and orthogonal compact manifolds, to obtain a more robust approximation for the ratio.

Significance. If the central derivation holds, the result would quantify a specific shortcoming of real constraints within complex-valued networks and supply an explicit asymptotic ratio. The nonstandard use of HCIZ for these volume comparisons could represent a methodological contribution to capacity calculations, but only if the analytic conditions for the integral representation are verified.

major comments (1)
  1. [HCIZ application in the main proof] The derivation of the headline asymptotic ratio (via Gardner-volume comparison at criticality) rests on an application of the HCIZ formula after Weyl reduction to the maximal torus. The manuscript provides no explicit check that the resulting effective potential satisfies the analyticity requirements or that the saddle-point contour remains valid once the real pre-activation constraint is imposed; this step is load-bearing for the claimed ratio.
minor comments (1)
  1. [Abstract] The abstract states that the HCIZ approach is 'nonstandard in literature'; a short paragraph contrasting the present usage with existing applications in statistical mechanics or random-matrix theory would help readers assess novelty.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their detailed review and for highlighting the need for explicit verification of analytic conditions in our application of the HCIZ formula. We address the single major comment below.

read point-by-point responses

  1. Referee: [HCIZ application in the main proof] The derivation of the headline asymptotic ratio (via Gardner-volume comparison at criticality) rests on an application of the HCIZ formula after Weyl reduction to the maximal torus. The manuscript provides no explicit check that the resulting effective potential satisfies the analyticity requirements or that the saddle-point contour remains valid once the real pre-activation constraint is imposed; this step is load-bearing for the claimed ratio.

    Authors: We agree that the manuscript does not currently contain an explicit verification that the effective potential remains analytic or that the saddle-point contour is valid after imposing the real pre-activation constraint. This verification is indeed necessary to rigorously justify the HCIZ application in the presence of the constraint. In the revised version we will add a dedicated appendix that establishes these conditions: we will show analyticity of the reduced potential on the maximal torus by direct differentiation under the integral sign (permitted by compactness of the unitary and orthogonal groups) and confirm contour validity by appealing to the absence of singularities inside the relevant contour for the Gardner volume at criticality, using the same Weyl reduction already employed in the main text. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation applies external HCIZ formula to Gardner volumes

full rationale

The paper derives the asymptotic storage capacity ratio by comparing Gardner volumes at criticality, using the HCIZ integral representation after Weyl reduction to the maximal torus with Haar measure over U(n) and O(n). The abstract explicitly frames this as an application of a known formula (Harish-Chandra-Itzykson-Zuber) that is external to the present work, with no equations or steps shown that reduce the claimed ratio to a fitted parameter, self-citation chain, or definitional tautology. No self-citations are referenced as load-bearing, and the method is described as nonstandard in application but not as importing uniqueness or ansatzes from prior author work. The derivation is therefore self-contained against the external integral identity.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract only; no explicit free parameters, axioms, or invented entities are detailed beyond the high-level reliance on Gardner volumes and the HCIZ formula.

pith-pipeline@v0.9.1-grok · 5622 in / 1122 out tokens · 34563 ms · 2026-06-28T06:50:55.241346+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

60 extracted references · 21 canonical work pages

  1. [1]

    A. B. Balantekin. Character expansions, itzykson-zuber integrals, and the qcd partition function. Physical Review D, 62(8), 2000. ISSN 1089-4918. doi: 10.1103/physrevd.62.085017. URL http: //dx.doi.org/10.1103/PhysRevD.62.085017

    work page doi:10.1103/physrevd.62.085017 2000

  2. [2]

    Malatesta, Gabriele Perugini, and Riccardo Zecchina

    Carlo Baldassi, Enrico M. Malatesta, Gabriele Perugini, and Riccardo Zecchina. Typical and atypical solutions in non-convex neural networks with discrete and continuous weights, 2023. URLhttps: //arxiv.org/abs/2304.13871

    arXiv 2023

  3. [3]

    Interpolating the sherrington–kirkpatrick replica trick.Philosophical Magazine, 92(1-3):78–97, January 2012

    Adriano Barra, Francesco Guerra, and Emanuele Mingione. Interpolating the sherrington–kirkpatrick replica trick.Philosophical Magazine, 92(1-3):78–97, January 2012. ISSN 1478-6443. doi: 10.1080/ 14786435.2011.637979. URLhttp://dx.doi.org/10.1080/14786435.2011.637979

    work page doi:10.1080/14786435.2011.637979 2012

  4. [4]

    J. A. Barrachina, C. Ren, G. Vieillard, C. Morisseau, and J.-P. Ovarlez. About the equivalence between complex-valued and real-valued fully connected neural networks - application to polinsar images. In 2021 IEEE 31st International Workshop on Machine Learning for Signal Processing (MLSP), pages 1–6,

    2021

  5. [5]

    doi: 10.1109/MLSP52302.2021.9596542

    work page doi:10.1109/mlsp52302.2021.9596542 2021

  6. [6]

    Complex-valued vs

    Jose Agustin Barrachina, Chenfang Ren, Christele Morisseau, Gilles Vieillard, and Jean-Philippe Ovarlez. Complex-valued vs. real-valued neural networks for classification perspectives: An example on non-circular data, 2021. URLhttps://arxiv.org/abs/2009.08340

    arXiv 2021

  7. [7]

    On guan’s examples of simply connected non-kahler compact complex manifolds

    Fedor A Bogomolov. On guan’s examples of simply connected non-kahler compact complex manifolds. American Journal of Mathematics, 118(5):1037–1046, 1996. doi: 10.1353/ajm.1996.0038

    work page doi:10.1353/ajm.1996.0038 1996

  8. [8]

    Lucas Böttcher and Mason A. Porter. Complex networks with complex weights.Physical Review E, 109,

  9. [9]

    doi: 10.1103/physreve.109.024314

    work page doi:10.1103/physreve.109.024314

  10. [10]

    Complex network for complex problems: A comparative study of cnn and complex-valued cnn

    Soumick Chatterjee, Pavan Tummala, Oliver Speck, and Andreas Nürnberger. Complex network for complex problems: A comparative study of cnn and complex-valued cnn. In2022 IEEE 5th International Conference on Image Processing Applications and Systems (IPAS), page 1–5. IEEE, December 2022. doi: 10.1109/ipas55744.2022.10053060. URLhttp://dx.doi.org/10.1109/IPA...

    work page doi:10.1109/ipas55744.2022.10053060 2022

  11. [11]

    Replica analysis of overfitting in regression models for time-to-event data.Journal of Physics A: Mathematical and Theoretical, 50(37):375001, August 2017

    A C C Coolen, J E Barrett, P Paga, and C J Perez-Vicente. Replica analysis of overfitting in regression models for time-to-event data.Journal of Physics A: Mathematical and Theoretical, 50(37):375001, August 2017. ISSN 1751-8121. doi: 10.1088/1751-8121/aa812f. URL http://dx.doi.org/10.1088/ 1751-8121/aa812f. 22

    work page doi:10.1088/1751-8121/aa812f 2017

  12. [12]

    On the capacity of neural networks, 2022

    Leonardo Cruciani. On the capacity of neural networks, 2022. URLhttps://arxiv.org/abs/2211. 07531

    2022

  13. [13]

    Identifying and attacking the saddle point problem in high-dimensional non-convex optimization, 2014

    Yann Dauphin, Razvan Pascanu, Caglar Gulcehre, Kyunghyun Cho, Surya Ganguli, and Yoshua Bengio. Identifying and attacking the saddle point problem in high-dimensional non-convex optimization, 2014. URLhttps://arxiv.org/abs/1406.2572

    Pith/arXiv arXiv 2014

  14. [14]

    Ricardo Estrada and Ram Kanwal.A Distributional Approach to Asymptotics. 01 2002. ISBN 978-1- 4612-6410-1. doi: 10.1007/978-0-8176-8130-2

    work page doi:10.1007/978-0-8176-8130-2 2002

  15. [15]

    The replica-symmetric free energy for ising spin glasses with orthogonally invariant couplings, 2024

    Zhou Fan and Yihong Wu. The replica-symmetric free energy for ising spin glasses with orthogonally invariant couplings, 2024. URLhttps://arxiv.org/abs/2105.02797

    arXiv 2024

  16. [16]

    Diffusion models and the manifold hypothesis: Log-domain smoothing is geometry adaptive, 2025

    Tyler Farghly, Peter Potaptchik, Samuel Howard, George Deligiannidis, and Jakiw Pidstrigach. Diffusion models and the manifold hypothesis: Log-domain smoothing is geometry adaptive, 2025. URLhttps: //arxiv.org/abs/2510.02305

    arXiv 2025

  17. [17]

    Forrester

    Peter J. Forrester. Meet andréief, bordeaux 1886, and andreev, kharkov 1882-83, 2018. URLhttps: //arxiv.org/abs/1806.10411

    Pith/arXiv arXiv 2018

  18. [18]

    E. Gardner. Maximum storage capacity in neural networks.Europhysics Letters, 4(4):481, aug 1987. doi: 10.1209/0295-5075/4/4/016. URLhttps://doi.org/10.1209/0295-5075/4/4/016

    work page doi:10.1209/0295-5075/4/4/016 1987

  19. [19]

    The space of interactions in neural network models.Journal of Physics A: Mathematical and General, 21(1):257, jan 1988

    E Gardner. The space of interactions in neural network models.Journal of Physics A: Mathematical and General, 21(1):257, jan 1988. doi: 10.1088/0305-4470/21/1/030. URLhttps://doi.org/10.1088/ 0305-4470/21/1/030

    work page doi:10.1088/0305-4470/21/1/030 1988

  20. [20]

    Gardner and Bernard Derrida

    E. Gardner and Bernard Derrida. Optimal storage properties of neural network models.Journal of Physics A: Mathematical and Theoretical, 21(1):271–284, 1988. doi: 10.1088/0305-4470/21/1/031. URL https://hal.science/hal-03285587

    work page doi:10.1088/0305-4470/21/1/031 1988

  21. [21]

    Sharp conditions for the bbm formula and asymptotics of heat content-type energies, 2025

    Luca Gennaioli and Giorgio Stefani. Sharp conditions for the bbm formula and asymptotics of heat content-type energies, 2025. URLhttps://arxiv.org/abs/2502.14655

    arXiv 2025

  22. [22]

    Wallach.Symmetry, Representations, and Invariants

    Roe Goodman and Nolan R. Wallach.Symmetry, Representations, and Invariants. Graduate Texts in Mathematics. Springer New York, 2009. doi: 10.1007/978-0-387-79852-3

    work page doi:10.1007/978-0-387-79852-3 2009

  23. [23]

    Hughes, J

    Jared A. Hughes, J. William Helton, and Peter Schlosser. The discrete laplace asymptotic method and its application to the 3xor satisfiability problem, 2025. URLhttps://arxiv.org/abs/2509.16420

    arXiv 2025

  24. [24]

    Spherical integrals of sublinear rank, 2023

    Jonathan Husson and Justin Ko. Spherical integrals of sublinear rank, 2023. URLhttps://arxiv.org/ abs/2208.03642

    arXiv 2023

  25. [25]

    Alan T. James. Zonal polynomials of the real positive definite symmetric matrices.Annals of Mathematics, 74(3):456–469, 1961. ISSN 0003486X, 19398980. URLhttp://www.jstor.org/stable/1970291

    arXiv 1961

  26. [26]

    Alan T. James. Distributions of matrix variates and latent roots derived from normal samples.Annals of Mathematical Statistics, 35:475–501, 1964. URLhttps://api.semanticscholar.org/CorpusID: 120381426

    1964

  27. [27]

    Inference from correlated patterns: a unified theory for perceptron learning and linear vector channels.Journal of Physics: Conference Series, 95:012001, January 2008

    Y Kabashima. Inference from correlated patterns: a unified theory for perceptron learning and linear vector channels.Journal of Physics: Conference Series, 95:012001, January 2008. ISSN 1742-6596. doi: 10.1088/1742-6596/95/1/012001. URLhttp://dx.doi.org/10.1088/1742-6596/95/1/012001

    work page doi:10.1088/1742-6596/95/1/012001 2008

  28. [28]

    Generalized random energy model at complex temperatures,

    Zakhar Kabluchko and Anton Klimovsky. Generalized random energy model at complex temperatures,

  29. [29]

    URLhttps://arxiv.org/abs/1402.2142

    Pith/arXiv arXiv

  30. [30]

    Kleinert

    H. Kleinert. Hubbard-stratonovich transformation: Successes, failure, and cure, 2011. URLhttps: //arxiv.org/abs/1104.5161. 23

    Pith/arXiv arXiv 2011

  31. [31]

    Fourier neural operator for parametric partial differential equations,

    Zongyi Li, Nikola Kovachki, Kamyar Azizzadenesheli, Burigede Liu, Kaushik Bhattacharya, Andrew Stuart, and Anima Anandkumar. Fourier neural operator for parametric partial differential equations,

  32. [32]

    URLhttps://arxiv.org/abs/2010.08895

    Pith/arXiv arXiv 2010

  33. [33]

    Terry A. Loring. The moore-osgood theorem on exchanging limits, 2010. URLhttps://math.unm.edu/ ~loring/links/analysis_f10/exchange.pdf

    2010

  34. [34]

    I. G. Macdonald.Symmetric Functions and Hall Polynomials. Oxford Mathematical Monographs. Oxford University Press, New York, 2nd edition, 1995

    1995

  35. [35]

    Antoine Maillard, Florent Krzakala, Marc Mézard, and Lenka Zdeborová. Perturbative construction of mean-field equations in extensive-rank matrix factorization and denoising.Journal of Statistical Mechanics: Theory and Experiment, 2022(8):083301, aug 2022. doi: 10.1088/1742-5468/ac7e4c. URL https://doi.org/10.1088/1742-5468/ac7e4c

    work page doi:10.1088/1742-5468/ac7e4c 2022

  36. [36]

    Malatesta

    Enrico M. Malatesta. High-dimensional manifold of solutions in neural networks: insights from statistical physics, 2025. URLhttps://arxiv.org/abs/2309.09240

    arXiv 2025

  37. [37]

    Muirhead

    Robb J. Muirhead. Aspects of multivariate statistical theory. InWiley Series in Probability and Statistics,

  38. [38]

    URLhttps://api.semanticscholar.org/CorpusID:123513635

  39. [39]

    Evaluation of complex-valued neural networks on real-valued classification tasks, 2018

    Nils Mönning and Suresh Manandhar. Evaluation of complex-valued neural networks on real-valued classification tasks, 2018. URLhttps://arxiv.org/abs/1811.12351

    Pith/arXiv arXiv 2018

  40. [40]

    Explicit generation of the branching tree of states in spin glasses.Journal of Statistical Mechanics: Theory and Experiment, 2015(5):P05002, May 2015

    G Parisi, F Ricci-Tersenghi, and D Yllanes. Explicit generation of the branching tree of states in spin glasses.Journal of Statistical Mechanics: Theory and Experiment, 2015(5):P05002, May 2015. ISSN 1742-5468. doi: 10.1088/1742-5468/2015/05/p05002. URL http://dx.doi.org/10.1088/1742-5468/ 2015/05/P05002

    work page doi:10.1088/1742-5468/2015/05/p05002 2015

  41. [41]

    An introduction to schur polynomials, 2018

    Amritanshu Prasad. An introduction to schur polynomials, 2018. URLhttps://arxiv.org/abs/1802. 06073

    2018

  42. [42]

    Generalizations of some integrals over the unitary group.Journal of Physics A: Mathematical and General, 36(12):3195–3201, March 2003

    B Schlittgen and T Wettig. Generalizations of some integrals over the unitary group.Journal of Physics A: Mathematical and General, 36(12):3195–3201, March 2003. ISSN 0305-4470. doi: 10.1088/0305-4470/ 36/12/319. URLhttp://dx.doi.org/10.1088/0305-4470/36/12/319

    work page doi:10.1088/0305-4470/ 2003

  43. [43]

    Hermann, Paris, 1966

    Laurent Schwartz.Théorie des distributions. Hermann, Paris, 1966

    1966

  44. [44]

    Rigorous solution of the gardner problem.Communications in Mathematical Physics, 234(3):383–422, mar 2003

    Mariya Shcherbina and Brunello Tirozzi. Rigorous solution of the gardner problem.Communications in Mathematical Physics, 234(3):383–422, mar 2003. doi: 10.1007/s00220-002-0783-3. URLhttps: //doi.org/10.1007/s00220-002-0783-3

    work page doi:10.1007/s00220-002-0783-3 2003

  45. [45]

    Pseudo-differential neural operator: Generalized fourier neural operator for learning solution operators of partial differential equations, 2024

    Jin Young Shin, Jae Yong Lee, and Hyung Ju Hwang. Pseudo-differential neural operator: Generalized fourier neural operator for learning solution operators of partial differential equations, 2024. URL https://arxiv.org/abs/2201.11967

    arXiv 2024

  46. [46]

    Perceptron capacity revisited: classification ability for correlated patterns.Journal of Physics A: Mathematical and Theoretical, 41(32):324013, 2008

    Takashi Shinzato and Yoshiyuki Kabashima. Perceptron capacity revisited: classification ability for correlated patterns.Journal of Physics A: Mathematical and Theoretical, 41(32):324013, 2008. ISSN 1751-8121. doi: 10.1088/1751-8113/41/32/324013. URL http://dx.doi.org/10.1088/1751-8113/41/ 32/324013

    work page doi:10.1088/1751-8113/41/32/324013 2008

  47. [47]

    Ben Sorscher, Robert Geirhos, Shashank Shekhar, Surya Ganguli, and Ari S. Morcos. Beyond neural scaling laws: beating power law scaling via data pruning. In Alice H. Oh, Alekh Agarwal, Danielle Belgrave, and Kyunghyun Cho, editors,Advances in Neural Information Processing Systems, 2022. URL https://openreview.net/forum?id=UmvSlP-PyV

    2022

  48. [48]

    On a weighted version of the bbm formula, 2025

    Giorgio Stefani. On a weighted version of the bbm formula, 2025. URLhttps://arxiv.org/abs/2504. 06736. 24

    2025

  49. [49]

    Fast ergodic search with kernel functions, 2025

    Max Muchen Sun, Ayush Gaggar, Peter Trautman, and Todd Murphey. Fast ergodic search with kernel functions, 2025. URLhttps://arxiv.org/abs/2403.01536

    arXiv 2025

  50. [50]

    The harish-chandra-itzykson-zuber integral formula, February 8 2013

    Terence Tao. The harish-chandra-itzykson-zuber integral formula, February 8 2013. URL https://terrytao.wordpress.com/2013/02/08/ the-harish-chandra-itzykson-zuber-integral-formula/

    2013

  51. [51]

    Storage capacity evaluation of the quantum perceptron using the replica method, 2024

    Mitsuru Urushibata and Masayuki Ohzeki. Storage capacity evaluation of the quantum perceptron using the replica method, 2024. URLhttps://arxiv.org/abs/2404.14785

    arXiv 2024

  52. [52]

    On the support of tempered distributions.Journal of Mathematical Analysis and Applications, 414(1):321–330, 2014

    Jasson Vindas and Ricardo Estrada. On the support of tempered distributions.Journal of Mathematical Analysis and Applications, 414(1):321–330, 2014. doi: 10.1016/j.jmaa.2014.01.009

    work page doi:10.1016/j.jmaa.2014.01.009 2014

  53. [53]

    An explicit formula for zonal polynomials, 2024

    Haoming Wang. An explicit formula for zonal polynomials, 2024. URLhttps://arxiv.org/abs/2410. 13558

    2024

  54. [54]

    Weisstein

    Eric W. Weisstein. Sifting property. From MathWorld—A Wolfram Resource. URLhttps://mathworld. wolfram.com/SiftingProperty.html

  55. [55]

    Princeton University Press, Princeton, NJ, 1940

    Hermann Weyl.The Classical Groups: Their Invariants and Representations. Princeton University Press, Princeton, NJ, 1940

    1940

  56. [56]

    Storage capacity of perceptron with variable selection, 2025

    Yingying Xu, Masayuki Ohzeki, and Yoshiyuki Kabashima. Storage capacity of perceptron with variable selection, 2025. URLhttps://arxiv.org/abs/2512.01861

    arXiv 2025

  57. [57]

    Spectral informed neural networks.Journal of Computational and Applied Mathematics, 477:117178, May 2026

    Tianchi Yu, Yiming Qi, Ivan Oseledets, and Shiyi Chen. Spectral informed neural networks.Journal of Computational and Applied Mathematics, 477:117178, May 2026. ISSN 0377-0427. doi: 10.1016/j.cam. 2025.117178. URLhttp://dx.doi.org/10.1016/j.cam.2025.117178

    work page doi:10.1016/j.cam 2026

  58. [58]

    Zavatone-Veth

    Jacob A. Zavatone-Veth. Expository notes on the pattern storage capacity of perceptrons with constrained weight distributions. Expository Notes, May 2025. Harvard University. Originally drafted June 2022, edited May 2025

    2025

  59. [59]

    William P. Ziemer. Modern real analysis.Graduate Texts in Mathematics, 2017. doi: 10.1007/ 978-3-319-64629-9. URL https://www.math.purdue.edu/~torresm/pubs/Modern-real-analysis. pdf

    2017

  60. [60]

    Łapiński

    Tomasz M. Łapiński. Approximations of the sum of states by laplace’s method for a system of particles with a finite number of energy levels and application to limit theorems.Mathematical Physics, Analysis and Geometry, 23(1), March 2020. ISSN 1572-9656. doi: 10.1007/s11040-020-9330-8. URL http://dx.doi.org/10.1007/s11040-020-9330-8. A Additional proofs A....

    work page doi:10.1007/s11040-020-9330-8 2020