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

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".