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Understanding neural networks with reproducing kernel Banach spaces

by F. Bartolucci and E. De Vito and L. Rosasco and S. Vigogna

(Report number 2021-30)

Abstract
Characterizing the function spaces corresponding to neural networks can provide a way to understand their properties. In this paper we discuss how the theory of reproducing kernel Banach spaces can be used to tackle this challenge. In particular, we prove a representer theorem for a wide class of reproducing kernel Banach spaces that admit a suitable integral representation and include one hidden layer neural networks of possibly infinite width. Further, we show that, for a suitable class of ReLU activation functions, the norm in the corresponding reproducing kernel Banach space can be characterized in terms of the inverse Radon transform of a bounded real measure, with norm given by the total variation norm of the measure. Our analysis simplifies and extends recent results in [34, 29, 30].

Keywords: neural networks, reproducing kernel Banach spaces, representer theorems, Radon transform

@Techreport{BDRV21_972,
  author = {F. Bartolucci and E. De Vito and L. Rosasco and S. Vigogna},
  title = {Understanding neural networks with reproducing kernel Banach spaces},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2021-30},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2021/2021-30.pdf },
  year = {2021}
}

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