Nonlinear frames and sparse reconstructions in Banach spaces

TL;DR

This paper extends frame theory to nonlinear maps between Banach spaces, establishing stable reconstruction algorithms and sparse signal recovery methods with convergence guarantees and new fixed point theorems.

Contribution

It introduces a nonlinear extension of frame theory using bi-Lipschitz maps and develops stable reconstruction algorithms with exponential convergence in Banach spaces.

Findings

Established exponential convergence of iterative algorithms for nonlinear measurements.

Developed a new fixed point theorem for well-localized maps on Banach spaces.

Demonstrated stable sparse signal recovery under the sparse Riesz and almost linear properties.

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 algorithm to reconstruct a signal $x$ from its noisy measurement $F(x)+\epsilon$ 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 later 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 ${\bf A}$ of closed linear subspaces of a Hilbert space ${\bf H}$ from their nonlinear measurements. We create an optimization framework called sparse approximation triple $({\bf A}, {\bf M}, {\bf H})$, and show that the minimizer $$x^*={\rm argmin}_{\hat x\in {\mathbf M}\ {\rm with} \ \|F(\hat x)-F(x^0)\|\le \epsilon} \|\hat x\|_{\mathbf M}$$ provides a suboptimal approximation to the original sparse signal $x^0\in {\bf A}$ when the measurement map $F$ has the sparse Riesz property and almost linear property on ${\mathbf A}$. The above two new properties is also discussed in this paper when $F$ is not far away from a linear measurement operator $T$ having the restricted isometry property.