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The stability of shallow neural networks on spheres: A sharp spectral analysis

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📄 Abstract

Abstract: We present an estimation of the condition numbers of the \emph{mass} and \emph{stiffness} matrices arising from shallow ReLU$^k$ neural networks defined on the unit sphere~$\mathbb{S}^d$. In particular, when $\{\theta_j^*\}_{j=1}^n \subset \mathbb{S}^d$ is \emph{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.

Key Contributions

This paper provides a sharp spectral analysis of shallow ReLU^k neural networks defined on spheres, estimating condition numbers of mass and stiffness matrices. It establishes a precise correspondence between approximation power and numerical stability, showing how eigenvalues relate to polynomial degrees, particularly for antipodally quasi-uniform data.

Business Value

Contributes to the fundamental understanding of neural network behavior, which can lead to the design of more stable and reliable models, especially for applications involving spherical or manifold data.

Paper Metadata

Innovation Type

Theoretical Analysis

Deployment Feasibility

Low, primarily theoretical research.

Limitations Addressed

Lack of theoretical understanding of neural network stability on non-Euclidean domains (spheres),Difficulty in characterizing the relationship between network architecture and its numerical properties,Understanding the impact of data distribution on network stability

Technical Tags

Shallow Neural NetworksSpheres (Manifolds)Spectral AnalysisCondition NumberMass MatrixStiffness MatrixReLU ActivationEigenvaluesEigenspacesNumerical StabilityApproximation PowerAntipodally Quasi-uniform

Research Topics

Neural Network TheoryGeometric Deep LearningNumerical AnalysisMachine Learning StabilityInterpretability

Methods & Architectures

Spectral AnalysisEigenvalue DecompositionCondition Number Estimation Shallow Neural Networks (ReLU^k)

Applications & Tasks

Machine Learning Theory Geometric Deep Learning Computer Vision (on spherical data) Data Analysis on Manifolds Analyzing Numerical StabilityUnderstanding Approximation PowerCharacterizing Network Behavior on Spheres Theoretical Analysis of Neural NetworksUnderstanding Network StabilityCharacterizing Network Expressivity

Related Fields

Machine Learning TheoryMathematicsNumerical AnalysisGeometryDeep Learning

Keywords

Neural NetworksStabilitySpheresSpectral AnalysisCondition NumberReLUEigenvaluesNumerical StabilityApproximationManifold LearningGeometric Deep Learning

Academic Context

#Neural Network Theory#Geometric Deep Learning#Numerical Analysis#Machine Learning Stability#Interpretability

Commercial Potential

Competitive Edge

Provides a rigorous theoretical framework for understanding neural network stability on spherical manifolds, advancing the field beyond Euclidean domains.

Resource Requirements

Compute Needs

Low, primarily analytical.

Data Requirements

Theoretical analysis, not data-dependent.

Deployment Constraints

N/A (theoretical paper).

Scalability

N/A (theoretical paper).

Production Readiness

Maturity Level

Theoretical Foundation

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