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This paper extends the concept of active subspaces from Euclidean domains to Riemannian manifolds by employing parallel transport to generalize the eigenvalue-ordered principle for analyzing scalar-valued quantities. The authors contrast intrinsic formulations with extrinsic, embedding-based approaches, revealing that while eigenvalues align to second order in geodesic radius, the dominant eigenspaces exhibit agreement relative to the spectral gap. Through numerical examples on the 2-sphere, the study demonstrates the practical implications of this framework, particularly in statistical shape analysis using hyperspheres.
Eigenvalues align in Riemannian manifolds, but the dominant eigenspaces reveal surprising differences that could redefine how we analyze shape data.
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.
