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Wikipedia / Algebraic signal processing

Algebraic signal processing (ASP) is an emerging area of theoretical signal processing (SP). In the algebraic theory of signal processing, a set of filterss is treated as an (abstract) algebra, a set of signalss is treated as a module or vector space, and convolution is treated as an algebra representation. The advantage of algebraic signal processing is its generality and portability.

History

-module.

Definitions

be a field is a vector space representing a set signals. A representation consists of an algebra homomorphism with composition (equivalent, in the finite-dimensional case, to matrix multiplication, convolution is called a shift operator. generated by shift operators. However, this is not necessarily the case for a representation of an arbitrary algebra.

Examples

Discrete signal processing

In discrete with bounded energy (i.e. square-integrable functions is the modulus of a complex number.

Graph signal processing

A weighted graph is an undirected graph with pseudometric. A graph signal is simply a real-valued function. There a few possible choices for a graph shift operator (GSO). The (un)normalized weighted adjacency matrix is a popular choice, as well as the (un)normalized graph Laplacian.

Other examples

Other mathematical objects with their own proposed signal-processing frameworks are algebraic signal models. These objects include including quivers, graphons, semilattices, finite groups, and Lie groups, and others.

Intertwining maps

In the framework of representation theory, relationships between two representations of the same algebra are described with intertwining maps. An intertwining map such that

. The z transform is a prototypical example of an intertwining map.

Algebraic neural networks

Inspired by a recent perspective that popular graph neural networks (GNNs) architectures are in fact convolutional neural networks (CNNs), recent work has been focused on developing novel neural network architectures from the algebraic point-of-view. An algebraic neural network is a composition of algebraic convolutions, possibly with multiple features and feature aggregations, and nonlinearities.

References