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 filters is treated as an (abstract) algebra, a set of signals 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

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In the original formulation of algebraic signal processing by Puschel and Moura, the signals are collected in an  -module for some algebra   of filters, and filtering is given by the action of   on the  -module.[1]

Definitions

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Let   be a field, for instance the complex numbers, and   be a  -algebra (i.e. a vector space over   with a binary operation   that is linear in both arguments) treated as a set of filters. Suppose   is a vector space representing a set signals. A representation of   consists of an algebra homomorphism   where   is the algebra of linear transformations   with composition (equivalent, in the finite-dimensional case, to matrix multiplication). For convenience, we write   for the endomorphism  . To be an algebra homomorphism,   must not only be a linear transformation, but also satisfy the property Given a signal  , convolution of the signal by a filter   yields a new signal  . Some additional terminology is needed from the representation theory of algebras. A subset   is said to generate the algebra if every element of   can be represented as polynomials in the elements of  . The image of a generator   is called a shift operator. In all practically all examples, convolutions are formed as polynomials in   generated by shift operators. However, this is not necessarily the case for a representation of an arbitrary algebra.

Examples

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Discrete Signal Processing

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In discrete signal processing (DSP), the signal space is the set of complex-valued functions   with bounded energy (i.e. square-integrable functions). This means the infinite series   where   is the modulus of a complex number. The shift operator is given by the linear endomorphism  . The filter space is the algebra of polynomials with complex coefficients   and convolution is given by   where   is an element of the algebra. Filtering a signal by  , then yields   because  .

Graph Signal Processing

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A weighted graph is an undirected graph   with pseudometric on the node set   written  . A graph signal is simply a real-valued function on the set of nodes of the graph. In graph neural networks, graph signals are sometimes called features. The signal space is the set of all graph signals   where   is a set of   nodes in  . The filter algebra is the algebra of polynomials in one indeterminate  . There a few possible choices for a graph shift operator (GSO). The (un)normalized weighted adjacency matrix of   is a popular choice, as well as the (un)normalized graph Laplacian  . The choice is dependent on performance and design considerations. If   is the GSO, then a graph convolution is the linear transformation   for some  , and convolution of a graph signal   by a filter   yields a new graph signal  .

Other Examples

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Other mathematical objects with their own proposed signal-processing frameworks are algebraic signal models. These objects include including quivers,[2] graphons,[3] semilattices,[4] finite groups, and Lie groups,[5] and others.

Intertwining Maps

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In the framework of representation theory, relationships between two representations of the same algebra are described with intertwining maps which in the context of signal processing translates to transformations of signals that respect the algebra structure. Suppose   and   are two different representations of  . An intertwining map is a linear transformation   such that

 

Intuitively, this means that filtering a signal by   then transforming it with   is equivalent to first transforming a signal with  , then filtering by  . The z transform[1] is a prototypical example of an intertwining map.

Algebraic Neural Networks

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Inspired by a recent perspective that popular graph neural networks (GNNs) architectures are in fact convolutional neural networks (CNNs),[6] recent work has been focused on developing novel neural network architectures from the algebraic point-of-view.[7][8] An algebraic neural network is a composition of algebraic convolutions, possibly with multiple features and feature aggregations, and nonlinearities.

References

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  1. ^ a b Puschel, M.; Moura, J. (2008). "Algebraic Signal Processing Theory: Foundation and 1-D Time". IEEE Transactions on Signal Processing. 56 (8): 3572–3585. Bibcode:2008ITSP...56.3572P. doi:10.1109/TSP.2008.925261. ISSN 1053-587X. S2CID 206797175.
  2. ^ Parada-Mayorga, Alejandro; Riess, Hans; Ribeiro, Alejandro; Ghrist, Robert (2020-10-22). "Quiver Signal Processing (QSP)". arXiv:2010.11525 [eess.SP].
  3. ^ Ruiz, Luana; Chamon, Luiz F. O.; Ribeiro, Alejandro (2021). "Graphon Signal Processing". IEEE Transactions on Signal Processing. 69: 4961–4976. arXiv:2003.05030. Bibcode:2021ITSP...69.4961R. doi:10.1109/TSP.2021.3106857. ISSN 1053-587X. S2CID 212657497.
  4. ^ Puschel, Markus; Seifert, Bastian; Wendler, Chris (2021). "Discrete Signal Processing on Meet/Join Lattices". IEEE Transactions on Signal Processing. 69: 3571–3584. arXiv:2012.04358. Bibcode:2021ITSP...69.3571P. doi:10.1109/TSP.2021.3081036. ISSN 1053-587X. S2CID 227736440.
  5. ^ Bernardini, Riccardo; Rinaldo, Roberto (2021). "Demystifying Lie Group Methods for Signal Processing: A Tutorial". IEEE Signal Processing Magazine. 38 (2): 45–64. Bibcode:2021ISPM...38b..45B. doi:10.1109/MSP.2020.3023540. ISSN 1053-5888. S2CID 232071730.
  6. ^ Gama, Fernando; Isufi, Elvin; Leus, Geert; Ribeiro, Alejandro (2020). "Graphs, Convolutions, and Neural Networks: From Graph Filters to Graph Neural Networks". IEEE Signal Processing Magazine. 37 (6): 128–138. arXiv:2003.03777. Bibcode:2020ISPM...37f.128G. doi:10.1109/MSP.2020.3016143. ISSN 1053-5888. S2CID 226292855.
  7. ^ Parada-Mayorga, Alejandro; Ribeiro, Alejandro (2021). "Algebraic Neural Networks: Stability to Deformations". IEEE Transactions on Signal Processing. 69: 3351–3366. arXiv:2009.01433. Bibcode:2021ITSP...69.3351P. doi:10.1109/TSP.2021.3084537. ISSN 1053-587X. S2CID 221517145.
  8. ^ Parada-Mayorga, Alejandro; Butler, Landon; Ribeiro, Alejandro (2023). "Convolutional Filtering and Neural Networks with Non Commutative Algebras". IEEE Transactions on Signal Processing. 71: 2683. arXiv:2108.09923. Bibcode:2023ITSP...71.2683P. doi:10.1109/TSP.2023.3293716.
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