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Approximation by non-symmetric networks for cross-domain learning.

H N Mhaskar1

  • 1Institute of Mathematical Sciences, Claremont Graduate University, Claremont, CA 91711, United States of America.

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Summary
This summary is machine-generated.

This study introduces a general method to analyze kernel-based networks with non-symmetric kernels, advancing machine learning approximation capabilities. It provides accuracy estimates for function approximation using ReLU networks, even with non-integer smoothness.

Keywords:
Cross-domain learningDegree of approximationNeural and kernel based approximation

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Area of Science:

  • Machine Learning
  • Approximation Theory
  • Neural Networks

Background:

  • Machine learning research has extensively studied the approximation capabilities (expressive power) of various models, including neural networks and kernel-based methods, for over three decades.
  • Existing research often relies on symmetric or positive definite kernels, limiting the scope of analysis for certain applications.

Purpose of the Study:

  • To develop a general framework for studying the approximation capabilities of kernel-based networks utilizing non-symmetric kernels.
  • To extend the analysis beyond traditional positive definite kernels, incorporating generalized translation networks and rotated zonal function kernels.
  • To obtain approximation accuracy estimates for functions in Sobolev classes using ReLU networks, particularly when the smoothness parameter 'r' is non-integer.

Main Methods:

  • Introduced a generalized approach to analyze kernel-based networks, moving beyond singular value decomposition for non-symmetric kernels.
  • Considered a family of kernels, including generalized translation networks and rotated zonal function kernels.
  • Derived uniform approximation accuracy estimates for functions in Sobolev classes using ReLU networks with non-integer smoothness parameters.

Main Results:

  • Established a general method for analyzing approximation capabilities of kernel-based networks with non-symmetric kernels.
  • Obtained specific accuracy estimates for uniform approximation of functions in Sobolev classes by ReLU networks, even for non-integer smoothness.
  • Demonstrated the applicability of the general results to functions with low smoothness relative to the input space dimension.

Conclusions:

  • The proposed general approach effectively analyzes kernel-based networks with non-symmetric kernels, expanding the theoretical understanding of their approximation power.
  • The findings provide valuable insights into the approximation accuracy of ReLU networks for functions with varying smoothness properties.
  • This work contributes to the theoretical foundations of machine learning, with potential implications for invariant learning, transfer learning, and advanced imaging techniques.