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

We study the fundamental limits of transferability inalgebraic signal processing through homomorphisms between algebraic signalmodels. Homomorphisms are linear maps between the signal spaces of twomodels that commute with filtering, so filtering a signal and transferringit across domains can be done in either order. The existence of such mapsis governed entirely by coincidences among the filtered eigenvalues of thetwo models' shift operators, but existence alone is insufficient: the spaceof homomorphisms always contains trivial elements that destroy allinformation. We introduce the spectral transfer efficiency$\eta(\theta)\in[0,1]$ to quantify information-preserving quality, provethat every homomorphism decomposes into unconstrained blocks overcoincidence classes, derive the dimension of the homomorphism space, andcharacterize exactly when lossless transfer is achievable. Beyond normalshift operators, we quantify a departure-from-normality penalty and showhow filter derivatives can repair spectral defectiveness. The theory yieldsconcrete consequences in three settings: for sampling, eigenvalueinterlacing converts transferability under subsampling into an explicitfilter design constraint; for compressed sensing, $\eta(\theta)$ controlsthe restricted isometry constant and the coherence of the resultingmeasurements, and recovery decouples across coincidence classes; and formachine learning, spectral aliasing emerges as the controlled symmetrybreaking that makes transfer between mismatched domains possible at all.