Hypothesis Spaces for Deep Learning

• URL: http://arxiv.org/abs/2403.03353v2
• Date: Mon, 11 Mar 2024 14:37:42 GMT
• Title: Hypothesis Spaces for Deep Learning
• Authors: Rui Wang, Yuesheng Xu, Mingsong Yan
• Abstract summary: This paper introduces a hypothesis space for deep learning that employs deep neural networks (DNNs) By treating a DNN as a function of two variables, we consider the primitive set of the DNNs for the parameter variable located in a set of the weight matrices and biases determined by a prescribed depth and widths of the DNNs. We prove that the Banach space so constructed is a kernel reproducing Banach space (RKBS) and construct its reproducing kernel.
• Score: 7.695772976072261
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: This paper introduces a hypothesis space for deep learning that employs deep neural networks (DNNs). By treating a DNN as a function of two variables, the physical variable and parameter variable, we consider the primitive set of the DNNs for the parameter variable located in a set of the weight matrices and biases determined by a prescribed depth and widths of the DNNs. We then complete the linear span of the primitive DNN set in a weak* topology to construct a Banach space of functions of the physical variable. We prove that the Banach space so constructed is a reproducing kernel Banach space (RKBS) and construct its reproducing kernel. We investigate two learning models, regularized learning and minimum interpolation problem in the resulting RKBS, by establishing representer theorems for solutions of the learning models. The representer theorems unfold that solutions of these learning models can be expressed as linear combination of a finite number of kernel sessions determined by given data and the reproducing kernel.

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