# Condition Numbers and Eigenvalue Spectra of Shallow Networks on Spheres

arXiv: https://arxiv.org/abs/2511.02625 · HTML: https://mlsanity.com/paper/2511.02625/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 24 of its 72 references that are in the map.
- First posted on arXiv 4 Nov 2025.

## Builds on (most cited first, 24 of 24)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Delving Deep into Rectifiers: Surpassing Human-Level Performance on ImageNet Classification](https://mlsanity.com/paper/1502.01852/) (2015)
- [Neural Tangent Kernel: Convergence and Generalization in Neural Networks](https://mlsanity.com/paper/1806.07572/) (2018)
- [On the Spectral Bias of Neural Networks](https://mlsanity.com/paper/1806.08734/) (2018)
- [The generalization error of random features regression: Precise asymptotics and double descent curve](https://mlsanity.com/paper/1908.05355/) (2019)
- [Frequency Principle: Fourier Analysis Sheds Light on Deep Neural Networks](https://mlsanity.com/paper/1901.06523/) (2019)
- [Breaking the Curse of Dimensionality with Convex Neural Networks](https://mlsanity.com/paper/1412.8690/) (2014)
- [Spectrum Dependent Learning Curves in Kernel Regression and Wide Neural Networks](https://mlsanity.com/paper/2002.02561/) (2020)
- [Two models of double descent for weak features](https://mlsanity.com/paper/1903.07571/) (2019)
- [Generalization Properties of Learning with Random Features](https://mlsanity.com/paper/1602.04474/) (2016)
- [On the Equivalence between Kernel Quadrature Rules and Random Feature Expansions](https://mlsanity.com/paper/1502.06800/) (2015)
- [Generalisation error in learning with random features and the hidden manifold model](https://mlsanity.com/paper/2002.09339/) (2020)
- [The Random Feature Model for Input-Output Maps between Banach Spaces](https://mlsanity.com/paper/2005.10224/) (2020)
- [Approximation by Combinations of ReLU and Squared ReLU Ridge Functions with  ℓ1  and  ℓ0  Controls](https://mlsanity.com/paper/1607.07819/) (2016)
- [Local Extreme Learning Machines and Domain Decomposition for Solving Linear and Nonlinear Partial Differential Equations](https://mlsanity.com/paper/2012.02895/) (2020)
- [Physics Informed Extreme Learning Machine (PIELM) -- A rapid method for the numerical solution of partial differential equations](https://mlsanity.com/paper/1907.03507/) (2019)
- [On the Activation Function Dependence of the Spectral Bias of Neural Networks](https://mlsanity.com/paper/2208.04924/) (2022)
- [Spectral Analysis of the Neural Tangent Kernel for Deep Residual Networks](https://mlsanity.com/paper/2104.03093/) (2021)
- [Randomized Nonlinear Component Analysis](https://mlsanity.com/paper/1402.0119/) (2014)
- [Deep Learning in Finance](https://mlsanity.com/paper/1602.06561/) (2016)
- [TransNet: Transferable Neural Networks for Partial Differential Equations](https://mlsanity.com/paper/2301.11701/) (2023)
- [Why Shallow Networks Struggle to Approximate and Learn High Frequencies](https://mlsanity.com/paper/2306.17301/) (2023)
- [Optimal Approximation of Zonoids and Uniform Approximation by Shallow Neural Networks](https://mlsanity.com/paper/2307.15285/) (2023)
- [A Multiple Transferable Neural Network Method with Domain Decomposition for Elliptic Interface Problems](https://mlsanity.com/paper/2502.19893/) (2025)

## Built on by (most cited first, 2 of 2)

- [Tight Worst-Case Bounds for the Smallest Eigenvalue of ReLU NTK Gram Matrices](https://mlsanity.com/paper/2608.03368/) (2026)
- [Dimension-Free Rank Lifting from Random Hyperplane Arrangements](https://mlsanity.com/paper/2609.39855/) (2026)

## Abstract (arXiv)

We present an estimation of the condition numbers of the mass and stiffness matrices arising from shallow ReLUk neural networks defined on the unit sphere~mathbbSd. In particular, when θj^*j=1n subset mathbbSd is antipodally quasi-uniform, the condition number is sharp. Indeed, in this case, we obtain sharp asymptotic estimates for the full spectrum of eigenvalues and characterize the structure of the corresponding eigenspaces, showing that the smallest eigenvalues are associated with an eigenbasis of low-degree polynomials while the largest eigenvalues are linked to high-degree polynomials. This spectral analysis establishes a precise correspondence between the approximation power of the network and its numerical stability.