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Geodesics in the Deep Linear Network

arXiv 2510.07324 ↗First posted 18 Sep 2025math.DG cs.LG math.DScited by 0

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Abstract (arXiv)

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 find horizontal straight lines in the invariant balanced manifold that remain geodesics under Riemannian submersion.

Key facts. No citing papers in the map yet (data through 14 Oct 2026). Builds on 6 of its 12 references that are in the map. First posted on arXiv 18 Sep 2025.

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References · 6

  1. 2018On the Optimization of Deep Networks: Implicit Acceleration by Overparameterizationcited by 284
  2. 2018A Convergence Analysis of Gradient Descent for Deep Linear Neural Networkscited by 184
  3. 2019Learning deep linear neural networks: Riemannian gradient flows and convergence to global minimizerscited by 61
  4. 2024The geometry of the deep linear networkcited by 11
  5. 2022Deep Linear Networks for Matrix Completion -- An Infinite Depth Limitcited by 8
  6. 2025An entropy formula for the Deep Linear Networkcited by 8

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