论文标题
内核PCA性能的上限和下限
Upper and Lower Bounds on the Performance of Kernel PCA
论文作者
论文摘要
主成分分析（PCA）是一种降低维度的流行方法，几十年来一直引起了人们的兴趣。最近，内核PCA（KPCA）已成为PCA的扩展，但是尽管它在实践中使用了，但仍缺少对KPCA的理论理解。我们对KPCA的效率贡献了几个下限和上限，涉及核革兰氏矩阵的经验特征值以及涉及方差概念的新数量。这些范围表明，KPCA平均捕获了多少信息，并对其效率有了更好的理论理解。我们证明，对于广泛使用的一类核，可以实现快速收敛速率，我们强调了数据集的某些理想特性以确保KPCA效率的重要性。
Principal Component Analysis (PCA) is a popular method for dimension reduction and has attracted an unfailing interest for decades. More recently, kernel PCA (KPCA) has emerged as an extension of PCA but, despite its use in practice, a sound theoretical understanding of KPCA is missing. We contribute several lower and upper bounds on the efficiency of KPCA, involving the empirical eigenvalues of the kernel Gram matrix and new quantities involving a notion of variance. These bounds show how much information is captured by KPCA on average and contribute a better theoretical understanding of its efficiency. We demonstrate that fast convergence rates are achievable for a widely used class of kernels and we highlight the importance of some desirable properties of datasets to ensure KPCA efficiency.
