  We investigate the connections between sparse approximation methods for
making kernel methods and Gaussian processes (GPs) scalable to large-scale
data, focusing on the Nystr\"om method and the Sparse Variational Gaussian
Processes (SVGP). While sparse approximation methods for GPs and kernel methods
share some algebraic similarities, the literature lacks a deep understanding of
how and why they are related. This may pose an obstacle to the communications
between the GP and kernel communities, making it difficult to transfer results
from one side to the other. Our motivation is to remove this obstacle, by
clarifying the connections between the sparse approximations for GPs and kernel
methods. In this work, we study the two popular approaches, the Nystr\"om and
SVGP approximations, in the context of a regression problem, and establish
various connections and equivalences between them. In particular, we provide an
RKHS interpretation of the SVGP approximation, and show that the Evidence Lower
Bound of the SVGP contains the objective function of the Nystr\"om
approximation, revealing the origin of the algebraic equivalence between the
two approaches. We also study recently established convergence results for the
SVGP and how they are related to the approximation quality of the Nystr\"om
method.
