Algebraic signal processing

In discrete signal processing (DSP), the signal space is the set of complex-valued functions ${\displaystyle {\mathcal {M}}={\mathcal {L}}^{2}(\mathbb {Z} )}$ with bounded energy (i.e. square-integrable functions). This means the infinite series ${\displaystyle \sum _{n=-\infty }^{\infty }|(x)_{n}|<\infty }$ where ${\displaystyle |\cdot |}$ is the modulus of a complex number. The shift operator is given by the linear endomorphism ${\displaystyle (Sx)_{n}=(x)_{n-1}}$. The filter space is the algebra of polynomials with complex coefficients ${\displaystyle {\mathcal {A}}=\mathbb {C} [z^{-1},z]}$ and convolution is given by ${\displaystyle \rho _{h}=\sum _{k=-\infty }^{\infty }h_{k}S^{k}}$ where ${\displaystyle h(t)=\sum _{k=-\infty }^{\infty }h_{k}z^{k}}$ is an element of the algebra. Filtering a signal by ${\displaystyle h}$, then yields ${\displaystyle (y)_{n}=\sum _{k=-\infty }^{\infty }h_{k}x_{n-k}}$ because ${\displaystyle (S^{k}x)_{n}=(x)_{n-k}}$.

A weighted graph is an undirected graph ${\displaystyle {\mathcal {G}}=({\mathcal {V}},{\mathcal {E}})}$ with pseudometric on the node set ${\displaystyle {\mathcal {V}}}$ written ${\displaystyle a_{ij}}$. 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 ${\displaystyle {\mathcal {M}}=\mathbb {R} ^{\mathcal {V}}}$ where ${\displaystyle {\mathcal {V}}}$ is a set of ${\displaystyle n=|{\mathcal {V}}|}$ nodes in ${\displaystyle {\mathcal {G}}=({\mathcal {V}},{\mathcal {E}})}$. The filter algebra is the algebra of polynomials in one indeterminate ${\displaystyle {\mathcal {A}}=\mathbb {R} [t]}$. There a few possible choices for a graph shift operator (GSO). The (un)normalized weighted adjacency matrix of ${\displaystyle [A]_{ij}=a_{ij}}$ is a popular choice, as well as the (un)normalized graph Laplacian ${\displaystyle [L]_{ij}={\begin{cases}\sum _{j=1}^{n}a_{ij}&i=j\\-a_{ij}&i\neq j\end{cases}}}$. The choice is dependent on performance and design considerations. If ${\displaystyle S}$ is the GSO, then a graph convolution is the linear transformation ${\displaystyle \rho _{h}=\sum _{k=0}^{\infty }h_{k}S^{k}}$ for some ${\displaystyle h(t)=\sum _{k=0}^{\infty }h_{k}z^{k}}$, and convolution of a graph signal ${\displaystyle \mathbf {x}$ :{\mathcal {V}}\to \mathbb {R} } by a filter ${\displaystyle h(t)}$ yields a new graph signal ${\displaystyle \mathbf {y} =\left(\sum _{k=0}^{\infty }h_{k}S^{k}\right)\cdot \mathbf {x} }$.

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.