This paper is concerned with the performance of Orthogonal Matching Pursuit (OMP) algorithms applied to a dictionary D in a Hilbert space H. Given an element f ∈ H, OMP generates a sequence of approximations fn, n = 1, 2, . . ., each of which is a linear combination of n dictionary elements chosen by a greedy criterion. It is studied whether the approximations fn are in some sense comparable to...

We show that ridgelets, a system introduced in [4], are optimal to represent smooth multivariate functions that may exhibit linear singularities. For instance, let {u · x− b > 0} be an arbitrary hyperplane and consider the singular function f(x) = 1{u·x−b>0}g(x), where g is compactly supported with finite Sobolev L2 norm ‖g‖Hs , s > 0. The ridgelet coefficient sequence of such an object is as s...

This paper is concerned with the performance of Orthogonal Matching Pursuit (OMP) algorithms applied to a dictionary D in a Hilbert space H. Given an element f ∈ H, OMP generates a sequence of approximations fn, n = 1, 2, . . ., each of which is a linear combination of n dictionary elements chosen by a greedy criterion. It is studied whether the approximations fn are in some sense comparable to...

We prove optimal embeddings for nonlinear approximation spaces Aq , in terms of weighted Lorentz sequence spaces, with the weights depending on the democracy functions of the basis. As applications we recover known embeddings for N -term wavelet approximation in L, Orlicz, and Lorentz norms. We also study the “greedy classes” G α q introduced by Gribonval and Nielsen, obtaining new counterexamp...

We investigate an N-particle Bose-Hubbard dimer with an additional effective decay term in one of the sites. A mean-field approximation for this non-Hermitian many-particle system is derived, based on a coherent state approximation. The resulting nonlinear, non-Hermitian two-level dynamics, in particular, the fixed point structures showing characteristic modifications of the self-trapping trans...

In compressed sensing we seek to gain information about vector x ∈ R from d << N nonadaptive linear measurements. Candes, Donoho, Tao et. al. ( see e.g. [2, 4, 8]) proposed to seek good approximation to x via `1 minimisation. In this paper we show that in the case of Gaussian measurements it recovers the signal well from inacurate measurements, thus improving result from [4]. We also show that ...

In this paper we show that the shearlets, an affine-like system of functions recently introduced by the authors and their collaborators, are essentially optimal in representing 2–dimensional functions f that are C2 except for discontinuities along C2 curves. More specifically, if fS N is the N–term reconstruction of f obtained by using the N largest coefficients in the shearlet representation, ...

This paper presents a new version of the High Dimensional Model Representation (HDMR) which attempts to approximate a given multivariate function by an expansion starting from a constant term and continuing by adding univariate components and then the terms whose multivariances increase via bivariance, trivariance and so on. HDMR works well as long as the function under consideration behaves, m...

This paper introduces a new Parseval frame, based on the 3–D shearlet representation, which is especially designed to capture geometric features such as discontinuous boundaries with very high efficiency. We show that this approach exhibits essentially optimal approximation properties for 3–D functions f which are smooth away from discontinuities along C2 surfaces. In fact, the N term approxima...