# Geodesics in the Deep Linear Network

arXiv: https://arxiv.org/abs/2510.07324 · HTML: https://mlsanity.com/paper/2510.07324/

## 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.

## Builds on (most cited first, 6 of 6)

- [On the Optimization of Deep Networks: Implicit Acceleration by Overparameterization](https://mlsanity.com/paper/1802.06509/) (2018)
- [A Convergence Analysis of Gradient Descent for Deep Linear Neural Networks](https://mlsanity.com/paper/1810.02281/) (2018)
- [Learning deep linear neural networks: Riemannian gradient flows and convergence to global minimizers](https://mlsanity.com/paper/1910.05505/) (2019)
- [The geometry of the deep linear network](https://mlsanity.com/paper/2411.09004/) (2024)
- [Deep Linear Networks for Matrix Completion -- An Infinite Depth Limit](https://mlsanity.com/paper/2210.12497/) (2022)
- [An entropy formula for the Deep Linear Network](https://mlsanity.com/paper/2509.09088/) (2025)

## 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.