分析某些贪心算法的一种统一方法
A unified way of analyzing some greedy algorithms
AI总结：
本文提出了一种统一分析Banach空间中特定类型的贪心算法的方法，定义了一类弱双正交贪心算法，涵盖了多种贪心算法，并探讨了其收敛性、收敛速率和数值稳定性。
摘要：
在本文中，我们提出了一种统一分析Banach空间中特定类型的贪心型算法的方法。我们定义了一类弱双正交贪心算法，其中包括广泛类型的贪心算法。特别地，我们证明了已知的弱切比雪夫贪心算法和弱贪心算法（自由放松）属于这一类。我们研究了弱双正交贪心算法的收敛性、收敛速率和数值稳定性。数值稳定性是指算法的步骤允许在受控的计算不准确性下执行。我们对那些不准确性幅度与算法收敛性质之间的联系进行了彻底分析。为了强调所提方法的优势，我们引入了从上述类中提出的新贪心算法——重标定的弱放松贪心算法，并在不显式分析算法的情况下推导了收敛结果。此外，我们解释了所提出的方法如何扩展到其他类型的贪心算法。
英文摘要：
In this paper we propose a unified way of analyzing a certain kind of greedy-type algorithms in Banach spaces. We define a class of the Weak Biorthogonal Greedy Algorithms that contains a wide range of greedy algorithms. In particular, we show that the following well-known algorithms --- the Weak Chebyshev Greedy Algorithm and the Weak Greedy Algorithm with Free Relaxation --- belong to this class. We investigate the properties of convergence, rate of convergence, and numerical stability of the Weak Biorthogonal Greedy Algorithms. Numerical stability is understood in the sense that the steps of the algorithm are allowed to be performed with controlled computational inaccuracies. We carry out a thorough analysis of the connection between the magnitude of those inaccuracies and the convergence properties of the algorithm. To emphasize the advantage of the proposed approach, we introduce here a new greedy algorithm --- the Rescaled Weak Relaxed Greedy Algorithm --- from the above class, and derive the convergence results without analyzing the algorithm explicitly. Additionally, we explain how the proposed approach can be extended to some other types of greedy algorithms.
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