An entropy formula for the Deep Linear Network
Abstract
We study the Riemannian geometry of the Deep Linear Network (DLN) as a foundation for a thermodynamic description of the learning process. The main tools are the use of group actions to analyze overparametrization and the use of Riemannian submersion from the space of parameters to the space of observables. The foliation of the balanced manifold in the parameter space by group orbits is used to define and compute a Boltzmann entropy. We also show that the Riemannian geometry on the space of observables defined in [2] is obtained by Riemannian submersion of the balanced manifold. The main technical step is an explicit construction of an orthonormal basis for the tangent space of the balanced manifold using the theory of Jacobi matrices.
Cite
@article{arxiv.2509.09088,
title = {An entropy formula for the Deep Linear Network},
author = {Govind Menon and Tianmin Yu},
journal= {arXiv preprint arXiv:2509.09088},
year = {2026}
}
Comments
Final version of accepted paper in SIAM Journal on Mathematical Analysis. Includes fixes of minor typos (especially equation (3.13), (6.35) and (6.36)