arXiv:2607.25163 2026 Geometry 1 ideas extracted · analyzed Aug 31, 2026
The paper turns active-subspace analysis into an intrinsic construction on a Riemannian manifold: gradients computed in different tangent spaces are parallel-transported to a common tangent space before forming an eigenvalue-ordered covariance. This is useful for neural networks whose inputs or latent states lie on normalized spheres, rotation manifolds, shape spaces, or other constrained representations, where averaging ambient Euclidean gradients can mix incompatible tangent directions. The most practical transfer is an intrinsic gradient-covariance module that discovers a small active frame and uses it to parameterize low-rank adapters, perturbations, or regularizers. The extracted hypothesis also supplies a falsifiable target: if a scalar network output is nearly a function of the learned active coordinates, restricting updates or inputs to that frame should preserve task performance while reducing trainable dimension or perturbation cost.
Learn a low-dimensional active frame for a neural scalar quantity on a curved latent manifold, rather than averaging gradients in unrelated ambient tangent spaces. Use the frame as the only input to a low-rank adapter or as a constraint on fine-tuning updates, with parallel transport making gradient statistics comparable across samples.