Nonlinear Frames and Sparse Reconstructions in Banach Spaces

作者：Sun Qiyu^*; Tang Wai Shing

来源：Journal of Fourier Analysis and Applications, 2017, 23(5): 1118-1152.

DOI：10.1007/s00041-016-9501-y

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

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 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 of closed linear subspaces of a Hilbert space from their nonlinear measurements. We introduce an optimization framework called a sparse approximation triple , and show that the minimizer
x* = argmin((x) over cap is an element of M) (with parallel to F((x) over cap)-F(x0)parallel to <=is an element of)parallel to(x) over cap parallel to M
provides a suboptimal approximation to the original sparse signal when the measurement map F has the sparse Riesz property and the almost linear property on . 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.

• 出版日期2017-10