The Geometry of the Set of Equivalent Linear Neural Networks

Abstract

We characterize the geometry of the set of all weight vectors for which a fully-connected linear neural network computes the same linear transformation $W$. Said another way, the set of all sequences of matrices (of predetermined sizes) whose product is $W$. This set of weight vectors is an algebraic variety in the Euclidean weight space, called the fiber of $W$ under the matrix multiplication map. Sometimes this variety is a manifold, but usually not. The rank stratification of a fiber is a natural partition of the fiber into smooth manifolds of various dimensions called strata. We derive the dimensions of these strata and characterize how they are connected to each other. They satisfy the frontier condition: if a stratum intersects the closure of another stratum, then the former stratum is a subset of the closure of the latter stratum. This subset relationship can be expressed as a partial order with a single minimal element, a stratum that is included in the closure of every other stratum. Hence the strata can be arranged in a hierarchy represented by a directed acyclic graph that represents that partial order; we give an algorithm that computes that graph. A key result describes the relationship between this partial order and the ranks of certain matrices in the network. We derive the tangent and normal subspaces of the strata, and we construct elegant bases for those subspaces. Each stratum represents a different pattern of information flow through the network, expressed as a barcode, borrowed from the study of persistent homology. The topology of a stratum depends solely on its barcode. So does its geometry, up to a linear transformation in weight space. Connections among the strata are best understood through "barcode moves," which can be expressed as simple modifications of the barcodes.