Hypothesis Spaces for Deep Learning

• URL: http://arxiv.org/abs/2403.03353v3
• Date: Thu, 14 Aug 2025 17:04:50 GMT
• Title: Hypothesis Spaces for Deep Learning
• Abstract summary: This paper introduces a hypothesis space for deep learning based on deep neural networks (DNNs)<n>By treating a DNN as a function of two variables - the input variable and the parameter variable - we consider the set of DNNs where the parameter variable belongs to a space of weight matrices and biases determined by a prescribed depth and layer widths.<n>We prove that the resulting Banach space is a reproducing kernel Banach space (RKBS) and explicitly construct its reproducing kernel.
• Score: 6.847383617201127
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: This paper introduces a hypothesis space for deep learning based on deep neural networks (DNNs). By treating a DNN as a function of two variables - the input variable and the parameter variable - we consider the set of DNNs where the parameter variable belongs to a space of weight matrices and biases determined by a prescribed depth and layer widths. To construct a Banach space of functions of the input variable, we take the weak* closure of the linear span of this DNN set. We prove that the resulting Banach space is a reproducing kernel Banach space (RKBS) and explicitly construct its reproducing kernel. Furthermore, we investigate two learning models - regularized learning and the minimum norm interpolation (MNI) problem - within the RKBS framework by establishing representer theorems. These theorems reveal that the solutions to these learning problems can be expressed as a finite sum of kernel expansions based on training data.

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