  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 Nystr\"om approximation
to the kernel matrix, the first such algorithm proposed. Incremental
calculation of the Nystr\"om approximation leads to further gains in memory
efficiency, and allows for empirical evaluation of when a subset of sufficient
size has been obtained.
