最大化输入空间中的边距
Maximing the Margin in the Input Space
摘要：
我们提出了一个新的标准支持向量机学习：最大化的输入空间，而不是在特征（希尔伯特）空间的边缘。该标准是Hastie等人提出的主曲线的判别版本。该标准特别适用于输入空间已经是具有相当小维度的精心设计的特征空间的情况。为了表示先验知识，对边缘的定义进行了推广。推导的算法包括两个交替的步骤来估计对偶参数。首先，初始化原始SVM的参数。然后，一组参数更新的牛顿类过程，另一组是通过解决一个二次规划问题更新。该算法在温和的条件下，几步收敛到局部最优，它保持了支持向量的稀疏性。虽然计算时间变量的复杂性增加，但求解二次规划问题的每一步的复杂性并没有改变。它还表明，原来的SVM可以被看作是一个特殊的情况。我们进一步推导出一个简化的算法，使我们能够使用现有的原始SVM的代码。
英文摘要：
We propose a novel criterion for support vector machine learning: maximizing the margin in the input space, not in the feature (Hilbert) space. This criterion is a discriminative version of the principal curve proposed by Hastie et al. The criterion is appropriate in particular when the input space is already a well-designed feature space with rather small dimensionality. The definition of the margin is generalized in order to represent prior knowledge. The derived algorithm consists of two alternating steps to estimate the dual parameters. Firstly, the parameters are initialized by the original SVM. Then one set of parameters is updated by Newton-like procedure, and the other set is updated by solving a quadratic programming problem. The algorithm converges in a few steps to a local optimum under mild conditions and it preserves the sparsity of support vectors. Although the complexity to calculate temporal variables increases the complexity to solve the quadratic programming problem for each step does not change. It is also shown that the original SVM can be seen as a special case. We further derive a simplified algorithm which enables us to use the existing code for the original SVM.
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