2023/06/01 by Willem Diepeveen, Diepeveen, Willem, Joyce Chew +3 · 1 citation
Mathematics · Medicine · #15A69 #49Q99 #53-04 #53-08 #53Z50 #90C26 #90C30 #Advanced Neuroimaging Techniques and Applications #Differential Geometry (math.DG) #FOS: Mathematics #Numerical Analysis (math.NA) #Optimization and Control (math.OC) #Tensor decomposition and applications
paper · pdf · doi:10.48550/arxiv.2306.00507
openalex publication_date 2023/06/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
When generalizing schemes for real-valued data approximation or decomposition to data living in Riemannian manifolds, tangent space-based schemes are very attractive for the simple reason that these spaces are linear. An open challenge is to do this in such a way that the generalized scheme is applicable to general Riemannian manifolds, is global-geometry aware and is computationally feasible. Existing schemes have been unable to account for all three of these key factors at the same time. In this work, we take a systematic approach to developing a framework that is able to account for all three factors. First, we will restrict ourselves to the -- still general -- class of symmetric Riemannian manifolds and show how curvature affects general manifold-valued tensor approximation schemes. Next, we show how the latter observations can be used in a general strategy for developing approximation schemes that are also global-geometry aware. Finally, having general applicability and global-geometry awareness taken into account we restrict ourselves once more in a case study on low-rank approximation. Here we show how computational feasibility can be achieved and propose the curvature-corrected truncated higher-order singular value decomposition (CC-tHOSVD), whose performance is subsequently tested in numerical experiments with both synthetic and real data living in symmetric Riemannian manifolds with both positive and negative curvature.