AI's Hidden Geometry: Riemannian Optimization on Manifolds (lightcapai.medium.com)

Researchers report a growing convergence between Riemannian optimization and manifold learning: instead of forcing problems into flat Euclidean space, modern methods treat constrained or intrinsically curved parameter/data spaces as bona fide manifolds and optimize on them. Practically this means using Riemannian gradients in tangent spaces, moving along geodesics or using retractions to map updates back to the manifold, and extending classic algorithms (gradient descent, conjugate gradient, trust-region, and stochastic gradient methods with momentum) to Stiefel, Grassmann, SPD, SO(3) and other manifolds. This approach naturally enforces constraints like orthonormality or positive-definiteness and often yields faster convergence and more accurate solutions than penalty-based Euclidean tricks. The convergence matters because many ML problems—and many data modalities—are inherently non-Euclidean: covariance matrices, rotations, low-rank subspaces, and learned low-dimensional data manifolds (Isomap, LLE, UMAP, t-SNE) benefit from geometry-aware treatment. Applications span orthogonal-weight neural layers, covariance-based classification, 3D vision/pose estimation, signal processing, and hyperbolic embeddings for representation learning. Accessible toolboxes (Manopt, pymanopt, Manopt.jl) are lowering the barrier to adoption, and ongoing research blending geometric optimization with statistical learning promises more robust, efficient algorithms; Riemannian methods are poised to become standard components of ML toolkits.