  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.
