Kernel methods are powerful learning methodologies that allow to perform non-linear data analysis. Despite their popularity, they suffer from poor scalability in big data scenarios. Various approximation methods, including random feature approximation, have been proposed to alleviate the problem. However, the statistical consistency of most of these approximate kernel methods is not well understood except for kernel ridge regression wherein it has been shown that the random feature approximation is not only computationally efficient but also statistically consistent with a minimax optimal rate of convergence. In this paper, we investigate the efficacy of random feature approximation in the context of kernel principal component analysis (KPCA) by studying the trade-off between computational and statistical behaviors of approximate KPCA. We show that the approximate KPCA is both computationally and statistically efficient compared to KPCA in terms of the error associated with reconstructing a kernel function based on its projection onto the corresponding eigenspaces. The analysis hinges on Bernstein-type inequalities for the operator and Hilbert-Schmidt norms of a self-adjoint Hilbert-Schmidt operator-valued U-statistics, which are of independent interest.

翻译：内核方法是能够进行非线性数据分析的强有力的学习方法。尽管广受欢迎,但在海量数据假设中,这些方法的伸缩性很差。提出了各种近似方法,包括随机地貌近似近似方法,以缓解问题。然而,大多数这类近似内核方法的统计一致性,除了内核脊柱回归之外,还没有得到很好理解,其中显示随机地貌近似方法不仅在计算上有效,而且在统计上也符合小型最佳汇合率。在本文中,我们通过研究近似金伯利公司计算和统计行为之间的取舍,调查随机地貌近似在主要组成部分分析(KPCA)中的效果。我们表明,从与KPCA有关的错误来看,KPA的大致在计算和统计上都是有效的,因为根据对相应电子空间的预测,重建内核内核函数的错误。分析取决于操作员伯恩斯坦式的不平等和Hilbert-Schmidt自接合的Hilbert-Sat-Sat-stat标准,这是独立的利益。