Convolutional Filtering with RKHS Algebras

TL;DR

This paper introduces a generalized convolutional framework within Reproducing Kernel Hilbert Spaces (RKHS), enabling scalable filtering and neural network models applicable to various data domains, including graphs and groups.

Contribution

It develops a novel algebraic convolutional theory for RKHS, allowing the construction of convolutional neural networks with explicit training expressions and broad applicability.

Findings

Enhanced neural network models in RKHS

Effective convolutional filtering on diverse data structures

Improved prediction accuracy in real-world wireless coverage data

Abstract

In this paper, we develop a generalized theory of convolutional signal processing and neural networks for Reproducing Kernel Hilbert Spaces (RKHS). Leveraging the theory of algebraic signal processing (ASP), we show that any RKHS allows the formal definition of multiple algebraic convolutional models. We show that any RKHS induces algebras whose elements determine convolutional operators acting on RKHS elements. This approach allows us to achieve scalable filtering and learning as a byproduct of the convolutional model, and simultaneously take advantage of the well-known benefits of processing information in an RKHS. To emphasize the generality and usefulness of our approach, we show how algebraic RKHS can be used to define convolutional signal models on groups, graphons, and traditional Euclidean signal spaces. Furthermore, using algebraic RKHS models, we build convolutional networks, formally defining the notion of pointwise nonlinearities and deriving explicit expressions for the training. Such derivations are obtained in terms of the algebraic representation of the RKHS. We present a set of numerical experiments on real data in which wireless coverage is predicted from measurements captured by unmaned aerial vehicles. This particular real-life scenario emphasizes the benefits of the convolutional RKHS models in neural networks compared to fully connected and standard convolutional operators.