  Symmetric positive semi-definite (SPSD) matrix approximation methods have
been extensively used to speed up large-scale eigenvalue computation and kernel
learning methods. The standard sketch based method, which we call the prototype
model, produces relatively accurate approximations, but is inefficient on large
square matrices. The Nystr\"om method is highly efficient, but can only achieve
low accuracy. In this paper we propose a novel model that we call the {\it fast
SPSD matrix approximation model}. The fast model is nearly as efficient as the
Nystr\"om method and as accurate as the prototype model. We show that the fast
model can potentially solve eigenvalue problems and kernel learning problems in
linear time with respect to the matrix size $n$ to achieve $1+\epsilon$
relative-error, whereas both the prototype model and the Nystr\"om method cost
at least quadratic time to attain comparable error bound. Empirical comparisons
among the prototype model, the Nystr\"om method, and our fast model demonstrate
the superiority of the fast model. We also contribute new understandings of the
Nystr\"om method. The Nystr\"om method is a special instance of our fast model
and is approximation to the prototype model. Our technique can be
straightforwardly applied to make the CUR matrix decomposition more efficiently
computed without much affecting the accuracy.
