# Geometry of fibers of the multiplication map of deep linear neural networks

arXiv: https://arxiv.org/abs/2411.19920 · HTML: https://mlsanity.com/paper/2411.19920/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 1 of its 48 references that are in the map.
- First posted on arXiv 29 Nov 2024.

## Builds on (most cited first, 1 of 1)

- [The Geometry of the Set of Equivalent Linear Neural Networks](https://mlsanity.com/paper/2404.14855/) (2024)

## Built on by (most cited first, 4 of 4)

- [From Global to Local: A Scalable Benchmark for Local Posterior Sampling](https://mlsanity.com/paper/2507.21449/) (2025)
- [Regularization Implies balancedness in the deep linear network](https://mlsanity.com/paper/2511.01137/) (2025)
- [A Basin-Selection Perspective on Grokking via Singular Learning Theory](https://mlsanity.com/paper/2603.01192/) (2026)
- [The Identifiability and Observability of Deep Normalized Attention](https://mlsanity.com/paper/2610.09620/) (2026)

## Abstract (arXiv)

We study the geometry of the algebraic set of tuples of composable matrices which multiply to a fixed matrix, using tools from the theory of quiver representations. In particular, we determine its codimension C and the number θ of its top-dimensional irreducible components. Our solution is presented in three forms: a Poincar'e series in equivariant cohomology, a quadratic integer program, and an explicit formula. In the course of the proof, we establish a surprising property: C and θ are invariant under arbitrary permutations of the dimension vector. We also show that the real log-canonical threshold of the function taking a tuple to the square Frobenius norm of its product is C/2. These results are motivated by the study of deep linear neural networks in machine learning and Bayesian statistics (singular learning theory) and show that deep linear networks are in a certain sense ``mildly singular".