Geodesics in the Deep Linear Network

Abstract:We derive a general system of ordinary differential equations and associated explicit solutions in a special case for geodesics between full rank matrices in the deep linear network geometry. In the process, we find horizontal straight lines in the invariant balanced manifold that remain geodesics under Riemannian submersion.

Differential Geometry (math.DG); Machine Learning (cs.LG); Dynamical Systems (math.DS)

arXiv:2510.07324 [math.DG]

(or arXiv:2510.07324v3 [math.DG] for this version)

https://doi.org/10.48550/arXiv.2510.07324 Focus to learn more arXiv-issued DOI via DataCite

arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website.

Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them.

Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.