Nonlinear Frames And Sparse Reconstructions In Banach Spaces

Keywords

Bi-Lipschitz property; Greedy algorithm; Nonlinear compressive sampling; Nonlinear frames; Restricted isometry property; Sparse approximation triple

Abstract

In the first part of this paper, we consider nonlinear extension of frame theory by introducing bi-Lipschitz maps F between Banach spaces. Our linear model of bi-Lipschitz maps is the analysis operator associated with Hilbert frames, p-frames, Banach frames, g-frames and fusion frames. In general Banach space setting, stable algorithms to reconstruct a signal x from its noisy measurement F(x) + ϵ may not exist. In this paper, we establish exponential convergence of two iterative reconstruction algorithms when F is not too far from some bounded below linear operator with bounded pseudo-inverse, and when F is a well-localized map between two Banach spaces with dense Hilbert subspaces. The crucial step to prove the latter conclusion is a novel fixed point theorem for a well-localized map on a Banach space. In the second part of this paper, we consider stable reconstruction of sparse signals in a union A of closed linear subspaces of a Hilbert space H from their nonlinear measurements. We introduce an optimization framework called a sparse approximation triple (A, M, H) , and show that the minimizer (Formula Presented.) provides a suboptimal approximation to the original sparse signal x0∈ A when the measurement map F has the sparse Riesz property and the almost linear property on A. The above two new properties are shown to be satisfied when F is not far away from a linear measurement operator T having the restricted isometry property.

Publication Date

10-1-2017

Publication Title

Journal of Fourier Analysis and Applications

Volume

23

Issue

5

Number of Pages

1118-1152

Document Type

Article

Personal Identifier

scopus

DOI Link

https://doi.org/10.1007/s00041-016-9501-y

Socpus ID

84988706142 (Scopus)

Source API URL

https://api.elsevier.com/content/abstract/scopus_id/84988706142

This document is currently not available here.

Share

COinS