aiqbench analysis
This research extends the active subspaces method—a key technique for dimensionality reduction in machine learning—from flat Euclidean spaces to curved Riemannian manifolds. By utilizing parallel transport, it offers an intrinsic formulation that contrasts with traditional embedding-based approaches, particularly relevant for statistical shape analysis and other geometric deep learning tasks. The study provides rigorous mathematical bounds on how these methods diverge when moving beyond local tangent approximations.
A Riemannian View on Active Subspaces
arXiv:2607.25163v1 Announce Type: cross Abstract: Active subspaces provide an explainable, eigenvalue-ordered principle for studying how scalar-valued quantities of interest change the most, on average, over a reduced basis of Euclidean domains. Composition with parallel transport generalizes this principle from Euclidean space to quantities of interest defined over Riemannian manifolds, and the resulting intrinsic formulation is contrasted with the extrinsic, embedding-based gradient average of manifold learning. Either strategy is studied in an intrinsically local sense, restricted to mean-centered geodesic-balls, and within that scope the two are not identical: on the central tangent space, eigenvalues agree to second order in the geodesic radius of the sampled domain, while dominant eigenspaces agree at the same order relative to the spectral gap. Extending activity beyond that central space then calls for either recomputed decompositions over changing tangent spaces or, intrinsically, parallel transport of a single central frame. Hyperspheres are emphasized throughout as a particular manifold of interest, motivated by applications over preshape spaces for statistical shape analysis. Numerical examples over the 2-sphere illustrate the formalism, including the derived ridge recovery at a curvature-limited quadratic rate.
Abstract:Active subspaces provide an explainable, eigenvalue-ordered principle for studying how scalar-valued quantities of interest change the most, on average, over a reduced basis of Euclidean domains. Composition with parallel transport generalizes this principle from Euclidean space to quantities of interest defined over Riemannian manifolds, and the resulting intrinsic formulation is contrasted with the extrinsic, embedding-based gradient average of manifold learning. Either strategy is studied in an intrinsically local sense, restricted to mean-centered geodesic-balls, and within that scope the two are not identical: on the central tangent space, eigenvalues agree to second order in the geodesic radius of the sampled domain, while dominant eigenspaces agree at the same order relative to the spectral gap. Extending activity beyond that central space then calls for either recomputed decompositions over changing tangent spaces or, intrinsically, parallel transport of a single central frame. Hyperspheres are emphasized throughout as a particular manifold of interest, motivated by applications over preshape spaces for statistical shape analysis. Numerical examples over the 2-sphere illustrate the formalism, including the derived ridge recovery at a curvature-limited quadratic rate.
Numerical Analysis (math.NA); Machine Learning (cs.LG); Differential Geometry (math.DG)
arXiv:2607.25163 [math.NA]
(or arXiv:2607.25163v1 [math.NA] for this version)
https://doi.org/10.48550/arXiv.2607.25163 Focus to learn more arXiv-issued DOI via DataCite (pending registration)
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