# On the Entropy Formula for Real, Complex, and Quaternionic Deep Linear Networks

arXiv: https://arxiv.org/abs/2606.16579 · HTML: https://mlsanity.com/paper/2606.16579/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 5 of its 10 references that are in the map.
- First posted on arXiv 15 Jun 2026.

## Builds on (most cited first, 5 of 5)

- [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)
- [Entropic Regularization in the Deep Linear Network](https://mlsanity.com/paper/2512.06137/) (2025)

## Abstract (arXiv)

We extend the entropy formula of Menon and Yu for the real Deep Linear Network (DLN) to its complex and quaternionic analogues, obtaining a unified formula for DLNs over ℝ, mathbbC, and mathbbH.