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On Neural Network Kernels and the Storage Capacity Problem.

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We found that the "effective order parameter" in statistical mechanics is the same as the Gaussian process kernel in infinite-width neural networks. This links storage capacity in treelike neural networks to kernel limits.

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Area of Science:

  • Machine Learning
  • Statistical Mechanics
  • Neural Networks

Background:

  • Wide neural networks are a key area of research.
  • Understanding their storage capacity and behavior in the infinite-width limit is crucial.
  • Existing literature explores these topics separately.

Purpose of the Study:

  • To connect the storage capacity problem in wide two-layer treelike neural networks with kernel limits in wide neural networks.
  • To establish a formal equivalence between key concepts from statistical mechanics and neural network theory.

Main Methods:

  • The study reifies the connection by observing the equivalence between the
  • effective order parameter
  • from statistical mechanics and the Gaussian process kernel in the infinite-width limit of neural networks.

Main Results:

  • The
  • effective order parameter
  • is shown to be exactly equivalent to the infinite-width neural network Gaussian process kernel.
  • This provides a bridge between statistical mechanics approaches to neural networks and kernel methods.

Conclusions:

  • The established correspondence unifies understanding of expressivity and trainability in wide two-layer treelike neural networks.
  • This finding facilitates cross-disciplinary insights between statistical mechanics and machine learning.