Limits and possibilities for transferring signals between algebraic models
Fundamental Limits of Transferability and Equivariance in Algebraic Signal Models I: Finite Dimensions
Machine Learning
Summary
This paper studies when and how signals, like sound or images, can be moved between different mathematical models without losing important information. The authors found that whether this transfer works depends on matching certain mathematical features called eigenvalues. They also created a way to measure how well information is preserved during transfer and identified when perfect transfer is possible. The results apply to areas like sampling signals, compressing data, and improving machine learning methods that use signals.
What this means in practice
- •For signal processing engineers: Design filters for subsampling that guarantee signal transferability by using eigenvalue interlacing constraints identified in this theory.
- •For machine learning engineers: Control and exploit spectral aliasing to enable machine learning models to transfer knowledge effectively between mismatched signal domains.
A theory result. No direct application yet.
Authors
Alejandro Parada-Mayorga
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.