An affine scaling methodology for best basis selection
An affine scaling methodology for best basis selection
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DOI：
10.1109/78.738251
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发表时间：
1999-01-01
影响因子：
5.4
通讯作者：
Kreutz-Delgado, K
中科院分区：
文献类型：
--
作者：
Rao, BD;Kreutz-Delgado, K
一种方法来推导算法的最佳基选择最小化的多样性措施提出的Wickerhauser和Donoho。这些度量包括类p范数（l（（p小于或等于1）多样性度量以及高斯熵和香农熵。算法开发方法使用因子表示的梯度，并涉及连续放松拉格朗日必要条件。这产生的算法是密切相关的仿射尺度变换（AST）的方法，通常采用的内点法非线性优化。最小化l（（p小于或等于1））多样性措施的算法是等价于最近开发的一类算法称为FOCal欠定系统求解器（FOCALS）。该方法的一般性质提供了一个系统的方法来推导这类算法和自然的机制来扩展它们。它还有助于更好地理解收敛行为和加强收敛结果。高斯熵最小化算法被证明是等价于一个良好的p = 0范数类优化算法。计算机实验表明，类p范数和高斯熵算法性能良好，收敛到稀疏解。香农熵算法产生的解决方案是集中的，但不收敛到一个完全稀疏的解决方案。
A methodology is developed to derive algorithms for optimal basis selection by minimizing diversity measures proposed by Wickerhauser and Donoho. These measures include the p-norm-like (l((p less than or equal to 1))) diversity measures and the Gaussian and Shannon entropies. The algorithm development methodology uses a factored representation for the gradient and involves successive relaxation of the Lagrangian necessary condition. This yields algorithms that are intimately related to the Affine Scaling Transformation (AST) based methods commonly employed by the interior point approach to nonlinear optimization. The algorithms minimizing the l((p less than or equal to 1)) diversity measures are equivalent to a recently developed class of algorithms called FOCal Underdetermined System Solver (FOCUSS). The general nature of the methodology provides a systematic approach for deriving this class of algorithms and a natural mechanism for extending them. It also facilitates a better understanding of the convergence behavior and a strengthening of the convergence results. The Gaussian entropy minimization algorithm is shown to be equivalent to a well-behaved p = 0 norm-like optimization algorithm. Computer experiments demonstrate that the p-norm-like and the Gaussian entropy algorithms perform well, converging to sparse solutions. The Shannon entropy algorithm produces solutions that are concentrated but are shown to not converge to a fully sparse solution.