Geodesics in the Deep Linear Network

Deep neural networks on manifold-valued representations have attracted growing interest, but many basic components remain tied to specific manifolds, rely on Euclidean approximations, or require costly and numerically fragile geometric operations. This thesis develops a unified framework for Riemannian deep learning from three complementary perspectives: reusable neural modules, manifold-specific network architectures, and the design of underlying geometries. It generalizes batch normalization from Euclidean spaces and individual manifolds to broad classes of Lie groups and gyrogroups, and extends multinomial logistic regression from Euclidean space to SPD manifolds and then to general Riemannian manifolds. It further develops neural networks for several important geometric representations, including an unconstrained model of hyperbolic space, Busemann-based hyperbolic learning, and full-rank correlation matrices. Finally, it introduces adaptive and computationally efficient Riemannian metrics on SPD manifolds, including learnable Log-Euclidean geometries and fast, stable Cholesky-based geometries. The proposed methods are supported by theoretical analysis and validated through numerical experiments and applications in vision, signal processing, graph learning, and genomics.

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.

We introduce a matrix generalization of the cone of Harvey and Lawson and prove that it is an exact special Lagrangian manifold. We further show that it belongs to a family of exact special Lagrangian manifolds that foliate the balanced manifold arising in deep learning.