Fredrik Hallgren

Fredrik Hallgren

The Nyström method is one of the most popular techniques for improving the scalability of kernel methods. However, it has not yet been derived for kernel PCA in line with classical PCA. In this paper we derive kernel PCA with the Nyström method, thereby providing one of the few available options to make kernel PCA scalable. We further study its statistical accuracy through a finite-sample confidence bound on the empirical reconstruction error compared to the full method. The behaviours of the method and bound are illustrated through computer experiments on multiple real-world datasets. As an application of the method we present kernel principal component regression with the Nyström method, as an alternative to Nyström kernel ridge regression for efficient regularized regression with kernels.

Fredrik Hallgren, Paul Northrop

Incremental versions of batch algorithms are often desired, for increased time efficiency in the streaming data setting, or increased memory efficiency in general. In this paper we present a novel algorithm for incremental kernel PCA, based on rank one updates to the eigendecomposition of the kernel matrix, which is more computationally efficient than comparable existing algorithms. We extend our algorithm to incremental calculation of the Nyström approximation to the kernel matrix, the first such algorithm proposed. Incremental calculation of the Nyström approximation leads to further gains in memory efficiency, and allows for empirical evaluation of when a subset of sufficient size has been obtained.