The elastic net: Stability for sparsity methods
The elastic net: Stability for sparsity methods
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弹性网络：稀疏方法的稳定性
DOI：
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发表时间：
2010
期刊：
影响因子：
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通讯作者：
S. Schiffler
中科院分区：
文献类型：
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作者：
S. Schiffler
近年来，稀疏逼近方法引起了人们的广泛关注，并成功地应用于各种数学学科中的许多问题。这项工作首先说明了稀疏近似的应用，从而引入了稀疏的概念。然后，介绍了该模型的数学框架和基本数学原理。特别是稀疏环境中的一个重要工具--1最小化，以及可用的算法。这为研究病态线性方程组的极小化稳定性问题提供了深刻的背景。事实证明，从统计学中产生的一种工具--弹性网络--承诺在减少稳定性问题的同时，保留1最小化的好处。讨论了1极小化与弹性网之间的关系。阐述了弹性网络的解析性质，并给出了相应的算法。对于样本问题，证明了1极小化的数值困难以及弹性网的影响。最后，在给定所有必要工具的情况下，讨论了弹性网极小化的精确恢复条件。
In recent years, methods for sparse approximation have gained considerable attention and have been successfully applied to numerous problems in various mathematical disciplines. This work starts by illustrating applications for sparse approximation to introduce the concept of sparsity. Afterwards, the mathematical framework and basic mathematical principles are introduced. Particularly 1 minimization, which is an important tool in the sparsity context, will be introduced as well as available algorithms. This forms a profound background to approach the problem of stability in 1 minimization for ill-conditioned linear equations. It turns out that a tool arising from statistics – the elastic net – promises to attenuate stability problems, while preserving the benefits of 1 minimization. The connection between 1 minimization and the elastic net is discussed. Analytical properties of the elastic net are stated and corresponding algorithms are developed. Numerical troubles of 1 minimization are demonstrated for sample problems as well as the influence of the elastic net. Finally, given all necessary tools, the discussion leads to the highlight of exact-recovery conditions for elastic-net minimization.