Nonlinear frames and sparse reconstructions in Banach spaces

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 $F(x)+ε$ 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 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 $$x^*={\rm argmin}_{\hat x\in {\mathbf M}\ {\rm with} \ \|F(\hat x)-F(x^0)\|\le ε} \|\hat x\|_{\mathbf M}$$ 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.