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Area of Science:

  • Machine Learning
  • Computer Vision
  • Theoretical Neuroscience

Background:

  • Fully-connected neural networks (infinite-width limit) often outperform finite-width counterparts in computer vision.
  • Modern convolutional neural network (CNN) architectures excel in the finite-width regime.

Purpose of the Study:

  • To provide a theoretical framework explaining performance differences between finite and infinite-width neural networks.
  • To analyze the behavior of convolutional layers versus fully-connected layers in shallow networks.

Main Methods:

  • Derivation of an effective action in the proportional limit for a one-hidden-layer convolutional network.
  • Comparison with the theoretical results for fully-connected networks.

Main Results:

  • Identified a distinct form of kernel renormalization in convolutional networks compared to fully-connected networks.
  • Convolutional kernels undergo local renormalization, enabling data-dependent selection of predictive components.
  • Fully-connected network kernels experience only global renormalization.

Conclusions:

  • Local kernel renormalization in CNNs provides a mechanism for feature learning in overparametrized shallow networks.
  • This feature learning capability is specific to CNNs and not present in shallow fully-connected or locally connected networks without weight sharing.