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Geometric multiscale decompositions of dynamic low-rank matrices
(2012)SAM Research ReportThe 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 ...Report -
Wolfowitz's theorem and consensus algorithms in Hadamard spaces
(2012)Research ReportWe provide a generalization ofWolfowitz’s theorem on the products of stochastic, indecomposable and aperiodic (SIA) matrices to metric spaces with nonpositive curvature. As a result we show convergence for a wide class of distributed consensus algorithms operating on these spaces.Report -
Hyperbolic cross approximation for the spatially homogeneous Boltzmann equation
(2012)Research ReportThe nonlinear spatially homogeneous integro-differential Boltzmann equation is a uniquely challenging task for numerical solvers due to the difficulty of efficiently computing the collision operator. A popular method is to expand the solution in Fourier modes and to truncate the collision operator. We present an approach based on the hyperbolic cross, whereby the performance can be greatly enhanced in some situations, as well as an offset ...Report -
Parabolic molecules
(2012)SAM Research ReportAnisotropic decompositions using representation systems based on parabolic scaling such as curvelets or shearlets have recently attracted significantly increased attention due to the fact that they were shown to provide optimally sparse approximations of functions exhibiting singularities on lower dimensional embedded manifolds. The literature now contains various direct proofs of this fact and of related sparse approximation results. ...Report -
Geometric multiscale decompositions of dynamic low-rank matrices
(2012)SAM Research ReportThe 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 ...Report -
Intrinsic Localization of Anisotropic Frames
(2012)SAM Research ReportThe present article studies o -diagonal decay properties of Moore-Penrose pseudoinverses of (bi-in finite) matrices satisfying an analogous condition. O -diagonal decay in our paper is considered with respect to speci c index distance functions which incorporates those usually used for the study of localization properties for wavelet frames but also more general systems such as curvelets or shearlets. Our main result is that if a matrix ...Report