Geodesics in the Deep Linear Network
This work addresses theoretical geometry problems in deep learning, but it appears incremental as it builds on existing mathematical frameworks without clear practical applications.
The authors derived a system of ordinary differential equations and explicit solutions for geodesics between full-rank matrices in deep linear network geometry, characterizing horizontal straight lines in the invariant balanced manifold that remain geodesics under Riemannian submersion.
We derive a general system of ODEs and associated explicit solutions in a special case for geodesics between full rank matrices in the deep linear network geometry. In the process, we characterize all horizontal straight lines in the invariant balanced manifold that remain geodesics under Riemannian submersion.