  In contrast with Mercer kernel-based approaches as used e.g., in Kernel
Principal Component Analysis (KPCA), it was previously shown that Singular
Value Decomposition (SVD) inherently relates to asymmetric kernels and
Asymmetric Kernel Singular Value Decomposition (KSVD) has been proposed.
However, the existing formulation to KSVD cannot work with infinite-dimensional
feature mappings, the variational objective can be unbounded, and needs further
numerical evaluation and exploration towards machine learning. In this work, i)
we introduce a new asymmetric learning paradigm based on coupled covariance
eigenproblem (CCE) through covariance operators, allowing infinite-dimensional
feature maps. The solution to CCE is ultimately obtained from the SVD of the
induced asymmetric kernel matrix, providing links to KSVD. ii) Starting from
the integral equations corresponding to a pair of coupled adjoint
eigenfunctions, we formalize the asymmetric Nystr\"om method through a finite
sample approximation to speed up training. iii) We provide the first empirical
evaluations verifying the practical utility and benefits of KSVD and compare
with methods resorting to symmetrization or linear SVD across multiple tasks.
