  We analyze the size of the dictionary constructed from online kernel
sparsification, using a novel formula that expresses the expected determinant
of the kernel Gram matrix in terms of the eigenvalues of the covariance
operator. Using this formula, we are able to connect the cardinality of the
dictionary with the eigen-decay of the covariance operator. In particular, we
show that under certain technical conditions, the size of the dictionary will
always grow sub-linearly in the number of data points, and, as a consequence,
the kernel linear regressor constructed from the resulting dictionary is
consistent.
