Convolutional Filtering with RKHS Algebras
This work provides a foundational framework for convolutional models in RKHS, potentially impacting machine learning and signal processing by offering a unified approach for diverse data types, though it appears incremental in extending existing algebraic methods.
The authors developed a generalized theory of convolutional signal processing and neural networks using Reproducing Kernel Hilbert Spaces (RKHS) and algebraic signal processing, enabling scalable filtering and learning across various domains like groups and graphons. They demonstrated its benefits in a real-world wireless coverage prediction task with UAV measurements, showing improved performance over standard methods.
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.