Using Composition Operators to Linearize LLM Semantic Transformations
Abstract
Machine learning learns functions: prompt to response, image to caption. What these functions are mathematically remains hard to say. We present a method to approximate these kinds of transformations using techniques from dynamical systems that fall under the umbrella of Koopmanism. We introduce the use of composition operators, which generalize the Koopman operator and, crucially, can map between distinct spaces, motivating the perspective that LLM transformations are rectangular infinite-dimensional operators. This formalism reveals useful structure: under natural assumptions on the prompt and response distributions, the LLM operator is an isometry, and misalignment between learned representations manifests as spectral pollution of its finite sections. We then outline a method of constructing finite-dimensional approximations of an LLM operator, and demonstrate how the singular value spectrum can be used to compare tasks and models.
Related
- Spherical harmonics, operators of multiplication by coordinates, and infinitesimal conformal transformations
- Weighted composition operators on the Fock space
- On the dynamics of composition operators: supercyclicity, odometers and translations
- Convexity of the Berezin range of composition operators induced by automorphisms of the Unit Ball in ℂn
- Weighted composition--differentiation operators on mixed-norm, derivative Hardy and F(p,q,s) spaces
BibTeX
@misc{chowdhury2026using,
title = {{Using Composition Operators to Linearize LLM Semantic Transformations}},
author = {Afjal Chowdhury and James Chen and Alan Edelman},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.22143},
eprint = {2609.22143},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.22143},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Using Composition Operators to Linearize LLM Semantic Transformations
AU - Afjal Chowdhury
AU - James Chen
AU - Alan Edelman
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.22143
UR - https://arxiv.org/abs/2609.22143
AB - Machine learning learns functions: prompt to response, image to caption. What these functions are mathematically remains hard to say. We present a method to approximate these kinds of transformations using techniques from dynamical systems that fall under the umbrella of Koopmanism. We introduce the use of composition operators, which generalize the Koopman operator and, crucially, can map between distinct spaces, motivating the perspective that LLM transformations are rectangular infinite-dimensional operators. This formalism reveals useful structure: under natural assumptions on the prompt and response distributions, the LLM operator is an isometry, and misalignment between learned representations manifests as spectral pollution of its finite sections. We then outline a method of constructing finite-dimensional approximations of an LLM operator, and demonstrate how the singular value spectrum can be used to compare tasks and models.
ER -