Sandra Keiper

Philipp Grohs, Sandra Keiper, Gitta Kutyniok et al.

Within the area of applied harmonic analysis, various multiscale systems such as wavelets, ridgelets, curvelets, and shearlets have been introduced and successfully applied. The key property of each of those systems are their (optimal) approximation properties in terms of the decay of the $L^2$-error of the best $N$-term approximation for a certain class of functions. In this paper, we introduce the general framework of $α$-molecules, which encompasses most multiscale systems from applied harmonic analysis, in particular, wavelets, ridgelets, curvelets, and shearlets as well as extensions of such with $α$ being a parameter measuring the degree of anisotropy, as a means to allow a unified treatment of approximation results within this area. Based on an $α$-scaled index distance, we first prove that two systems of $α$-molecules are almost orthogonal. This leads to a general methodology to transfer approximation results within this framework, provided that certain consistency and time-frequency localization conditions of the involved systems of $α$-molecules are satisfied. We finally utilize these results to enable the derivation of optimal sparse approximation results \msch{for} a specific class of cartoon-like functions by sufficient conditions on the 'control' parameters of a system of $α$-molecules.

Axel Flinth, Sandra Keiper

This work treats the recovery of sparse, binary signals through box-constrained basis pursuit using biased measurement matrices. Using a probabilistic model, we provide conditions under which the recovery of both sparse and saturated binary signals is very likely. In fact, we also show that under the same condition, the solution of the boxed-constrained basis pursuit program can be found using boxed-constrained least-squares.

Sandra Keiper

This paper studies the approximation of generalized ridge functions, namely of functions which are constant along some submanifolds of $\mathbb{R}^N$. We introduce the notion of linear-sleeve functions, whose function values only depend on the distance to some unknown linear subspace $L$. We propose two effective algorithms to approximate linear-sleeve functions $f(x)=g(\text{dist}(x,L)^2)$, when both the linear subspace $L\subset \mathbb{R}^N$ and the function $g\in C^s[0,1]$ are unknown. We will prove error bounds for both algorithms and provide an extensive numerical comparison of both. We further propose an approach of how to apply these algorithms to capture general sleeve functions, which are constant along some lower dimensional submanifolds.