Fundamental Limits of Transferability and Equivariance in Algebraic Signal Models I: Finite Dimensions

Alejandro Parada-Mayorga

eess.SPSep 28, 2026

Fundamental Limits of Transferability and Equivariance in Algebraic Signal Models I: Finite Dimensions

Authors: Alejandro Parada-Mayorga

Organizations: Department of Electrical Engineering, University of Colorado (Denver)

Abstract

We study the fundamental limits of transferability in algebraic signal processing through homomorphisms between algebraic signal models. Homomorphisms are linear maps between the signal spaces of two models that commute with filtering, so filtering a signal and transferring it across domains can be done in either order. The existence of such maps is governed entirely by coincidences among the filtered eigenvalues of the two models' shift operators, but existence alone is insufficient: the space of homomorphisms always contains trivial elements that destroy all information. We introduce the spectral transfer efficiency $η(θ)\in[0,1]$ to quantify information-preserving quality, prove that every homomorphism decomposes into unconstrained blocks over coincidence classes, derive the dimension of the homomorphism space, and characterize exactly when lossless transfer is achievable. Beyond normal shift operators, we quantify a departure-from-normality penalty and show how filter derivatives can repair spectral defectiveness. The theory yields concrete consequences in three settings: for sampling, eigenvalue interlacing converts transferability under subsampling into an explicit filter design constraint; for compressed sensing, $η(θ)$ controls the restricted isometry constant and the coherence of the resulting measurements, and recovery decouples across coincidence classes; and for machine learning, spectral aliasing emerges as the controlled symmetry breaking that makes transfer between mismatched domains possible at all.

Figures & tables

Panel of Figure 1 — Figure 1: Homomorphism between algebraic signal models. The linear map $\theta:{\mathcal{H}}_{1}\to{\mathcal{H}}_{2}$ makes the diagram commute for every filter $a\in{\mathcal{A}}$ : filtering in $({\mathcal{A}},{\mathcal{H}}_{1},\rho_{1})$ and then transferring with $\theta$ produces the same signal as transferring first and then filtering in $({\mathcal{A}},{\mathcal{H}}_{2},\rho_{2})$ .

Panel of Figure 2 — Figure 2: Coincidence classes arise from equality of the filtered eigenvalues, with or without flatness of the filter. Left: A filter with a flat plateau: every eigenvalue falling inside the plateau is mapped to the same value, so the coincidence class is produced by the flatness itself. Right: A non-flat, non-monotonic filter: aliasing here arises purely from $p(t)$ having two local maxima, since the level $c_{1}$ is crossed once ascending and once descending on the first hump, and again while ascending into the second. The eigenvalues $\lambda_{1}$ and $\lambda_{2}$ of $\mathbf{S}$ (navy dots), located near entirely different extrema, still satisfy $p(\lambda_{1})=p(\lambda_{2})=c_{1}$ , while a single eigenvalue of $\widetilde{\mathbf{S}}$ (gold triangle) on the descending edge of the first hump completes the class ( $m_{c_{1}}=2$ , $\widetilde{m}_{c_{1}}=1$ ). The level $c_{2}$ , reached only near the taller second hump, gives an ordinary one-to-one coincidence ( $m_{c_{2}}=\widetilde{m}_{c_{2}}=1$ ). An eigenvalue of $\mathbf{S}$ past the second hump is annihilated, as its filtered image matches no eigenvalue of $\widetilde{\mathbf{S}}$ .

Panel of Figure 3 — Figure 3: Geometric picture of the spectral transfer efficiency $\eta_{c}(\theta)$ : the unit sphere of $\mathcal{E}_{c}$ (dashed) mapped by $\theta_{c}$ into its image (filled) in $\widetilde{\mathcal{E}}_{c}$ . Left ( $\eta_{c}(\theta)=1$ ): a scaled isometry maps the sphere onto a uniformly shrunk circle – every direction has the same gain $\sigma_{\min}(\theta_{c})=\sigma_{\max}(\theta_{c})$ , and transfer is lossless. Middle ( $0<\eta_{c}(\theta)<1$ ): the image is an ellipse with distinct semi-axes $\sigma_{\max}(\theta_{c}),\sigma_{\min}(\theta_{c})$ , so gains differ and the signal is distorted. Right ( $\eta_{c}(\theta)=0$ ): the image collapses to a segment, $\sigma_{\min}(\theta_{c})=0$ , and that direction’s information is destroyed before reaching $\widetilde{\mathcal{E}}_{c}$ .

Panel of Figure 4 — Figure 4: Cauchy interlacing as a design constraint under node subsampling. Navy dots are eigenvalues of $\mathbf{S}$ ( $N_{1}=6$ nodes) and gold triangles are two eigenvalues of $\widetilde{\mathbf{S}}$ after removing $N_{1}-N_{2}=2$ nodes. Interlacing guarantees $\widetilde{\lambda}_{i}\in[\lambda_{i+N_{1}-N_{2}},\lambda_{i}]$ regardless of which nodes are removed, shown shaded for $i=1,4$ . Making $p(t)$ constant on each bracket guarantees $p(\widetilde{\lambda}_{i})=p(\lambda_{i})$ independently of the sampling set. Elsewhere, $p$ retains full spectral resolution.

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