On Quantile Regression in Reproducing Kernel Hilbert Spaces with the Data Sparsity Constraint
On Quantile Regression in Reproducing Kernel Hilbert Spaces with the Data Sparsity Constraint
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发表时间：
2016
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通讯作者：
Chong Zhang;Yufeng Liu;Yichao Wu
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作者：
Chong Zhang;Yufeng Liu;Yichao Wu
对于样条回归，众所周知，节点的选择对估计器的性能至关重要。作为覆盖光滑样条的通用学习框架，再生核希尔伯特空间（RKHS）中的学习也有类似的问题。然而，在RKHS表示核函数的训练数据点的选择还没有仔细研究的文献。在本文中，我们研究了分位数回归作为RKHS中学习的一个例子。在这种情况下，正则平方范数惩罚不执行训练数据选择。我们提出了一个数据稀疏约束，对核函数系数进行阈值处理，以实现稀疏核函数表示。我们证明了所提出的数据稀疏性方法在某些情况下可以具有竞争力的预测性能，并且在其他情况下与传统的平方范数惩罚相比具有相当的性能。因此，数据稀疏方法可以作为一个有竞争力的替代平方范数惩罚方法。我们提出的方法使用数据稀疏性约束的一些理论性质。模拟和真实的数据集被用来证明我们的数据稀疏约束的有用性。
For spline regressions, it is well known that the choice of knots is crucial for the performance of the estimator. As a general learning framework covering the smoothing splines, learning in a Reproducing Kernel Hilbert Space (RKHS) has a similar issue. However, the selection of training data points for kernel functions in the RKHS representation has not been carefully studied in the literature. In this paper we study quantile regression as an example of learning in a RKHS. In this case, the regular squared norm penalty does not perform training data selection. We propose a data sparsity constraint that imposes thresholding on the kernel function coefficients to achieve a sparse kernel function representation. We demonstrate that the proposed data sparsity method can have competitive prediction performance for certain situations, and have comparable performance in other cases compared to that of the traditional squared norm penalty. Therefore, the data sparsity method can serve as a competitive alternative to the squared norm penalty method. Some theoretical properties of our proposed method using the data sparsity constraint are obtained. Both simulated and real data sets are used to demonstrate the usefulness of our data sparsity constraint.