L1最小化的动态更新
Dynamic Updating for L1 Minimization
摘要：
压缩感知（CS）理论表明，在一定条件下，稀疏信号可以从少量的线性非相干测量恢复。一类有效的重建算法涉及求解凸优化程序，该程序平衡解的L1范数与数据保真度项。近年来，求解这些L1最小化程序的算法取得了巨大的进展。然而，这些算法在很大程度上是静态的：它们专注于为一组固定的测量值找到解决方案。在本文中，我们将讨论“动态算法”解决L1最小化程序流集的测量。我们考虑的情况下，潜在的信号之间的测量略有变化，并在一个固定的信号的新的测量顺序添加到系统。我们开发的算法，以快速更新几种不同类型的L1优化问题的解决方案，每当这些变化发生，从而避免从头开始解决一个新的优化问题。我们提出的方案是基于同伦连续，分解成一个小数目的线性步骤的解决方案更新系统和有效的方式。每一步都由一个低秩更新和少量的矩阵向量乘法组成--非常类似于递归最小二乘法。我们的调查还包括L1解码问题的动态更新计划，其中任意信号是从冗余编码测量已被稀疏错误损坏恢复。
英文摘要：
The theory of compressive sensing (CS) suggests that under certain conditions, a sparse signal can be recovered from a small number of linear incoherent measurements. An effective class of reconstruction algorithms involve solving a convex optimization program that balances the L1 norm of the solution against a data fidelity term. Tremendous progress has been made in recent years on algorithms for solving these L1 minimization programs. These algorithms, however, are for the most part static: they focus on finding the solution for a fixed set of measurements. In this paper, we will discuss "dynamic algorithms" for solving L1 minimization programs for streaming sets of measurements. We consider cases where the underlying signal changes slightly between measurements, and where new measurements of a fixed signal are sequentially added to the system. We develop algorithms to quickly update the solution of several different types of L1 optimization problems whenever these changes occur, thus avoiding having to solve a new optimization problem from scratch. Our proposed schemes are based on homotopy continuation, which breaks down the solution update in a systematic and efficient way into a small number of linear steps. Each step consists of a low-rank update and a small number of matrix-vector multiplications -- very much like recursive least squares. Our investigation also includes dynamic updating schemes for L1 decoding problems, where an arbitrary signal is to be recovered from redundant coded measurements which have been corrupted by sparse errors.
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