基于L1-Homotopy的流信号稀疏恢复
Sparse Recovery of Streaming Signals Using L1-Homotopy
摘要：
大多数现有的稀疏信号恢复方法假设一个静态系统：未知信号是一个有限长度的向量，其中一组固定的线性测量和稀疏表示的基础是可用的，并解决了L1范数最小化程序的重建。然而，相同的表示和重构框架并不容易应用于流式传输系统中：未知信号随时间变化，并且在小的时间间隔内顺序地测量和重构。在本文中，我们讨论了两个这样的流式系统和同伦算法快速解决相关的L1范数最小化程序：1）恢复一个光滑的，时变信号，而不是使用块变换，我们使用重叠正交变换的稀疏表示。2)恢复遵循线性动态模型的稀疏时变信号。对于这两个系统，我们迭代处理测量的滑动区间和估计稀疏系数通过求解加权L1范数最小化程序。而不是从头开始解决一个新的L1程序在每次迭代，我们使用一个可用的信号估计作为一个同伦公式的起点。从一个热启动向量开始，我们的同伦算法随着系统的变化以少量计算成本低廉的步骤更新解决方案。本文提出的同伦算法是高度通用的，因为它可以更新的L1问题的解决方案，在一些动态设置。我们用数值实验证明，我们提出的流恢复框架优于表示和重建信号作为独立的，不相交的块的方法，在重建质量方面，我们提出的基于同伦的更新方案优于当前国家的最先进的求解器的计算时间和复杂性。
英文摘要：
Most of the existing methods for sparse signal recovery assume a static system: the unknown signal is a finite-length vector for which a fixed set of linear measurements and a sparse representation basis are available and an L1-norm minimization program is solved for the reconstruction. However, the same representation and reconstruction framework is not readily applicable in a streaming system: the unknown signal changes over time, and it is measured and reconstructed sequentially over small time intervals. In this paper, we discuss two such streaming systems and a homotopy-based algorithm for quickly solving the associated L1-norm minimization programs: 1) Recovery of a smooth, time-varying signal for which, instead of using block transforms, we use lapped orthogonal transforms for sparse representation. 2) Recovery of a sparse, time-varying signal that follows a linear dynamic model. For both the systems, we iteratively process measurements over a sliding interval and estimate sparse coefficients by solving a weighted L1-norm minimization program. Instead of solving a new L1 program from scratch at every iteration, we use an available signal estimate as a starting point in a homotopy formulation. Starting with a warm-start vector, our homotopy algorithm updates the solution in a small number of computationally inexpensive steps as the system changes. The homotopy algorithm presented in this paper is highly versatile as it can update the solution for the L1 problem in a number of dynamical settings. We demonstrate with numerical experiments that our proposed streaming recovery framework outperforms the methods that represent and reconstruct a signal as independent, disjoint blocks, in terms of quality of reconstruction, and that our proposed homotopy-based updating scheme outperforms current state-of-the-art solvers in terms of the computation time and complexity.
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