Banach空间中的非线性框架与稀疏重建
Nonlinear frames and sparse reconstructions in Banach spaces
AI总结：
本文研究Banach空间中非线性框架的扩展及其在信号重建中的应用，提出稳定重建算法并探讨稀疏信号的优化框架。
摘要：
本文第一部分考虑通过引入Banach空间间的双 Lipschitz 映射 F 扩展框架理论。我们的线性模型包括与 Hilbert 框架、p-框架、Banach 框架、g-框架和融合框架相关的分析算子。在一般Banach空间中，从噪声测量 F(x)+ε 重建信号 x 可能不存在稳定算法。本文证明当 F 接近某个有下界线性算子且具有有界伪逆时，两种迭代重建算法具有指数收敛性。当 F 是两个Banach空间之间的良好局部化映射且具有稠密Hilbert子空间时，证明的关键步骤是为Banach空间上的良好局部化映射提出的新固定点定理。第二部分研究在Hilbert空间H的闭合线性子空间联盟A中，从非线性测量重建稀疏信号。提出稀疏逼近三元组（A, M, H）的优化框架，并证明当测量映射F具有稀疏Riesz性质和近线性性质时，最小化问题的解提供原始稀疏信号x0的次优近似。本文还讨论当F接近具有受限等距性质的线性测量算子T时，上述两种新性质的情况。
英文摘要：
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.
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