Consistency and robustness of kernel-based regression in convex risk minimization
Consistency and robustness of kernel-based regression in convex risk minimization
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DOI：
10.3150/07-bej5102
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发表时间：
2007-08-01
期刊：
影响因子：
1.5
通讯作者：
Steinwart, Ingo
中科院分区：
文献类型：
--
作者：
Christmann, Andreas;Steinwart, Ingo
我们研究了一类广泛的现代基于核的回归（KBR）方法的统计特性。这些核方法是在过去的十年中发展起来的，其灵感来自于无穷维希尔伯特空间中的凸风险最小化。一个主要的例子是支持向量回归。我们首先描述KBR方法的损失函数L和响应变量的尾部之间的关系。然后，我们建立了KBR的L-风险的一致性，给出了数学上的理由，这些方法能够“学习”的声明。然后，我们考虑这样的核方法的鲁棒性。特别是，我们的研究结果使我们能够选择损失函数和内核，以获得计算上易于处理和一致的KBR方法，有界的影响函数。此外，界限的偏差和灵敏度曲线，这是一个有限样本版本的影响函数，和KBR和经典的M估计之间的关系进行了讨论。
We investigate statistical properties for a broad class of modern kernel-based regression (KBR) methods. These kernel methods were developed during the last decade and are inspired by convex risk minimization in infinite-dimensional Hilbert spaces. One leading example is support vector regression. We first describe the relationship between the loss function L of the KBR method and the tail of the response variable. We then establish the L-risk consistency for KBR which gives the mathematical justification for the statement that these methods are able to "learn". Then we consider robustness properties of such kernel methods. In particular, our results allow us to choose the loss function and the kernel to obtain computationally tractable and consistent KBR methods that have bounded influence functions. Furthermore, bounds for the bias and for the sensitivity curve, which is a finite sample version of the influence function, are developed, and the relationship between KBR and classical M estimators is discussed.