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

TL;DR

This paper investigates the geometric structure of the set of matrix tuples forming a fixed product, revealing invariances and properties relevant to deep linear neural networks and singular learning theory.

Contribution

It provides explicit formulas and invariance properties for the geometry of matrix multiplication fibers, connecting algebraic geometry with deep learning.

Findings

Determines codimension and number of top-dimensional components of the matrix tuple set.

Establishes invariance of these geometric properties under permutations of dimensions.

Links the real log-canonical threshold to the codimension, informing singular learning theory.

Abstract

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 $\theta$ 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 $\theta$ 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".