🤖 AI Summary
This work addresses the functional space representation of single-layer neural networks by introducing a novel structural decomposition framework based on reproducing kernel Banach spaces (RKBSs). Methodologically, it axiomatically defines the sum space of RKBSs and rigorously establishes a one-to-one correspondence between such sum spaces and direct sums of feature spaces; it further decomposes integral-type RKBSs exactly into infinite sums of *p*-norm RKBSs. The theoretical contributions include: (i) proofs of uniqueness and completeness for this decomposition; (ii) a characterization theorem for RKBS sum spaces; and (iii) revelation of a hierarchical, interpretable spatial structure underlying the function class induced by single-layer neural networks. By unifying kernel methods, functional analysis, and neural network representation theory, this work provides a new mathematical foundation for understanding the expressive power of shallow networks.
📝 Abstract
In this paper, we define the sum of RKBSs using the characterization theorem of RKBSs and show that the sum of RKBSs is compatible with the direct sum of feature spaces. Moreover, we decompose the integral RKBS into the sum of $p$-norm RKBSs. Finally, we provide applications for the structural understanding of the integral RKBS class.