In this article, we introduce the concept of samplets by transferring the construction of Tausch-White wavelets to the realm of data. This way we obtain a multilevel representation of discrete data which directly enables data compression, detection of singularities and adaptivity. Applying samplets to represent kernel matrices, as they arise in kernel based learning or Gaussian process regression, we end up with quasi-sparse matrices. By thresholding small entries, these matrices are compressible to O(N log N) relevant entries, where N is the number of data points. This feature allows for the use of fill-in reducing reorderings to obtain a sparse factorization of the compressed matrices. Besides the comprehensive introduction to samplets and their properties, we present extensive numerical studies to benchmark the approach. Our results demonstrate that samplets mark a considerable step in the direction of making large data sets accessible for analysis.
翻译:在本篇文章中,我们引入样本概念,将Tausch-White波子的构造转移到数据领域。 这样我们就能获得不同数据的多层代表, 从而直接实现数据压缩、 发现奇点和适应性。 应用样本代表内核矩阵, 当它们出现在内核学习或高斯进程回归中时, 我们最终会出现准零散的矩阵。 通过起始小条目, 这些矩阵可以压缩到O( Nlog N) 相关条目中, 其中N是数据点的数量。 这个特征允许使用填入减少重新排序以获得压缩矩阵的稀少因子化。 除了对样本及其特性的全面介绍外, 我们提出广泛的数字研究来为该方法的基准基准。 我们的结果显示, 样本标志着在为分析提供大数据集方面迈出了相当长的一步。