arxiv_stat_ml 70% Match Research Paper Machine learning theorists,Deep learning researchers,Mathematicians 2 months ago

Does the Barron space really defy the curse of dimensionality?

reinforcement-learning › robotics-rl

📄 Abstract

Abstract: The Barron space has become famous in the theory of (shallow) neural networks because it seemingly defies the curse of dimensionality. And while the Barron space (and generalizations) indeed defies (defy) the curse of dimensionality from the POV of classical smoothness, we herein provide some evidence in favor of the idea that the Barron space (and generalizations) does (do) not defy the curse of dimensionality with a nonclassical notion of smoothness which relates naturally to "infinitely wide" shallow neural networks. Like how the Bessel potential spaces are defined via the Fourier transform, we define so-called ADZ spaces via the Mellin transform; these ADZ spaces encapsulate the nonclassical smoothness we alluded to earlier. 38 pages, will appear in the dissertation of the author

Key Contributions

This paper provides evidence that the Barron space, while defying the curse of dimensionality in a classical sense, does not defy it with a nonclassical notion of smoothness related to infinitely wide shallow neural networks. It introduces ADZ spaces, defined via the Mellin transform, to capture this nonclassical smoothness.

Business Value

Deepens the theoretical understanding of neural network expressivity and generalization, potentially leading to the design of more efficient and powerful network architectures.

Paper Metadata

Innovation Type

Theoretical Framework

Deployment Feasibility

Theoretical; implications for algorithm design.

Limitations Addressed

The common understanding that the Barron space completely defies the curse of dimensionality, proposing a more nuanced view based on different notions of smoothness.

Technical Tags

Barron spacecurse of dimensionalityneural networkssmoothnessMellin transformADZ spacesfunction approximationmachine learning theory

Research Topics

Neural Network TheoryCurse of DimensionalityFunction ApproximationSmoothness SpacesFourier Analysis

Methods & Architectures

Mellin transformdefinition of ADZ spacesanalysis of smoothness Shallow Neural Networks

Applications & Tasks

Machine Learning Theory Deep Learning Investigating whether the Barron space truly defies the curse of dimensionalityDefining new notions of smoothness related to infinitely wide neural networks Function approximationUnderstanding neural network expressivity

Related Fields

Machine Learning TheoryDeep LearningFunctional AnalysisHarmonic AnalysisNumerical Analysis

Keywords

Barron spacecurse of dimensionalityneural networkssmoothnessMellin transformfunction approximationdeep learning theoryexpressivityADZ spacesFourier analysis

Academic Context

#Neural Network Theory#Curse of Dimensionality#Function Approximation#Smoothness Spaces#Fourier Analysis

Commercial Potential

Target Industries

TechnologyResearch

Competitive Edge

Offers a refined perspective on the Barron space's properties concerning the curse of dimensionality, introducing new mathematical tools (ADZ spaces).

Market Opportunity

Fundamental research impacting AI.

Revenue Models

N/A

Resource Requirements

Compute Needs

Low (theoretical analysis)

Data Requirements

N/A

Deployment Constraints

Theoretical implications, not direct deployment.

Scalability

Relates to the scalability of neural network expressivity with width.

Production Readiness

Maturity Level

Theoretical

Time to Market

Long (impact on future research)

Patent Potential

Low

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