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

arXiv: https://arxiv.org/abs/2609.36106 · HTML: https://mlsanity.com/paper/2609.36106/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 19 of its 41 references that are in the map.
- First posted on arXiv 28 Sep 2026.

## Builds on (most cited first, 19 of 19)

- [Scalars are universal: Equivariant machine learning, structured like classical physics](https://mlsanity.com/paper/2106.06610/) (2021)
- [Algebraic Neural Networks: Stability to Deformations](https://mlsanity.com/paper/2009.01433/) (2020)
- [Graphon Filters: Graph Signal Processing in the Limit](https://mlsanity.com/paper/2003.02099/) (2020)
- [Machine learning and invariant theory](https://mlsanity.com/paper/2209.14991/) (2022)
- [Causal Fourier Analysis on Directed Acyclic Graphs and Posets](https://mlsanity.com/paper/2209.07970/) (2022)
- [Graphon Pooling in Graph Neural Networks](https://mlsanity.com/paper/2003.01795/) (2020)
- [The Graphon Fourier Transform](https://mlsanity.com/paper/1910.10195/) (2019)
- [Convolutional Learning on Multigraphs](https://mlsanity.com/paper/2209.11354/) (2022)
- [Discrete Signal Processing with Set Functions](https://mlsanity.com/paper/2001.10290/) (2020)
- [Stability of Algebraic Neural Networks to Small Perturbations](https://mlsanity.com/paper/2010.11544/) (2020)
- [Discrete Signal Processing on Meet/Join Lattices](https://mlsanity.com/paper/2012.04358/) (2020)
- [Convolutional Filtering and Neural Networks with Non Commutative Algebras](https://mlsanity.com/paper/2108.09923/) (2021)
- [Graphon Pooling for Reducing Dimensionality of Signals and Convolutional Operators on Graphs](https://mlsanity.com/paper/2212.08171/) (2022)
- [Lie Group Algebra Convolutional Filters](https://mlsanity.com/paper/2305.04431/) (2023)
- [Learning with Multigraph Convolutional Filters](https://mlsanity.com/paper/2210.16272/) (2022)
- [Sampling and Uniqueness Sets in Graphon Signal Processing](https://mlsanity.com/paper/2401.06279/) (2024)
- [Algebraic Convolutional Filters on Lie Group Algebras](https://mlsanity.com/paper/2210.17425/) (2022)
- [Convolutional Filtering with RKHS Algebras](https://mlsanity.com/paper/2411.01341/) (2024)
- [RKHS Representation of Algebraic Convolutional Filters with Integral Operators](https://mlsanity.com/paper/2602.19094/) (2026)

## Abstract (arXiv)

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)∈[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.