Report 2012-03

Geometric multiscale decompositions of dynamic low-rank matrices

P. Grohs

Abstract: The present paper is concerned with the study of manifold-valued multiscale transforms with a focus on the Stiefel manifold. For this specic 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.

Keywords: Riemannian data, low-rank approximation, N-term approximation, compression, manifold-valued wavelet transforms.

Paper: Available as PDF (1,1MB) or as hardcopy to order reports@sam.math.ethz.ch.