An introduction to wavelets
The Geometry of Algorithms with Orthogonality Constraints
SIAM Journal on Matrix Analysis and Applications
On Smooth Decompositions of Matrices
SIAM Journal on Matrix Analysis and Applications
Smoothness and Periodicity of Some Matrix Decompositions
SIAM Journal on Matrix Analysis and Applications
Singular value decomposition of time-varying matrices
Future Generation Computer Systems - Special issue: Geometric numerical algorithms
A Dual Approach to Semidefinite Least-Squares Problems
SIAM Journal on Matrix Analysis and Applications
Dynamical Low-Rank Approximation
SIAM Journal on Matrix Analysis and Applications
Dynamical low-rank approximation: applications and numerical experiments
Mathematics and Computers in Simulation
Optimization Algorithms on Matrix Manifolds
Optimization Algorithms on Matrix Manifolds
Approximation order from stability for nonlinear subdivision schemes
Journal of Approximation Theory
Dynamical Tensor Approximation
SIAM Journal on Matrix Analysis and Applications
The Undecimated Wavelet Decomposition and its Reconstruction
IEEE Transactions on Image Processing
Projection-like Retractions on Matrix Manifolds
SIAM Journal on Optimization
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The present paper is concerned with the study of manifold-valued multiscale transforms with a focus on the Stiefel manifold. For this specific geometry we derive several formulas and algorithms for the computation of geometric means which will later enable us to construct multiscale transforms of wavelet type. As an application we study compression of piecewise smooth families of low-rank matrices both for synthetic data and also real-world data arising in hyperspectral imaging. As a main theoretical contribution we show that the manifold-valued wavelet transforms can achieve an optimal N-term approximation rate for piecewise smooth functions with possible discontinuities. This latter result is valid for arbitrary manifolds.