We analyze the Nystrom approximation of a positive definite kernel associated with a probability measure. We first prove an improved error bound for the conventional Nystrom approximation with i.i.d. sampling and singular-value decomposition in the continuous regime; the proof techniques are borrowed from statistical learning theory. We further introduce a refined selection of subspaces in Nystrom approximation with theoretical guarantees that is applicable to non-i.i.d. landmark points. Finally…