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) + 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 A of closed linear subspaces of a Hilbert space H from their nonlinear measurements. We create an optimization framework called sparse approximation triple (A,M,H), and show that the minimizer x = argminx̂∈M with ‖F (x̂)−F (x0)‖≤ ‖x̂‖M provides a suboptimal approximation to the original sparse signal x ∈ A when the measurement map F has the sparse Riesz property and almost linear property on 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.