用于内核LMS的在线字典学习。分析和前向-后向分裂算法
Online dictionary learning for kernel LMS. Analysis and forward-backward splitting algorithm
摘要：
自适应滤波算法在再生核Hilbert空间中的非线性系统辨识中表现出优于线性滤波算法的优点。不幸的是，这些方法的一个不期望的特性是滤波器的阶数随着输入数据的数量线性增长。这大大增加了计算负担和内存需求。已经提出了各种基于字典学习的策略来克服这个严重的缺点。很少，如果有的话，这些作品分析的问题，更新字典在一个随时间变化的环境。在本文中，我们提出了一个分析研究的高斯最小均方算法的收敛行为的情况下，字典元素的统计只有部分匹配的输入数据的统计。这使我们能够强调以在线方式更新字典的必要性，丢弃过时的元素并添加适当的元素。我们引入了一个核最小均方算法与L1范数正则化自动执行这项任务。分析了该方法的平均稳定性，并通过实验验证了其性能。
英文摘要：
Adaptive filtering algorithms operating in reproducing kernel Hilbert spaces have demonstrated superiority over their linear counterpart for nonlinear system identification. Unfortunately, an undesirable characteristic of these methods is that the order of the filters grows linearly with the number of input data. This dramatically increases the computational burden and memory requirement. A variety of strategies based on dictionary learning have been proposed to overcome this severe drawback. Few, if any, of these works analyze the problem of updating the dictionary in a time-varying environment. In this paper, we present an analytical study of the convergence behavior of the Gaussian least-mean-square algorithm in the case where the statistics of the dictionary elements only partially match the statistics of the input data. This allows us to emphasize the need for updating the dictionary in an online way, by discarding the obsolete elements and adding appropriate ones. We introduce a kernel least-mean-square algorithm with L1-norm regularization to automatically perform this task. The stability in the mean of this method is analyzed, and its performance is tested with experiments.
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