  Many unsupervised kernel methods rely on the estimation of the kernel
covariance operator (kernel CO) or kernel cross-covariance operator (kernel
CCO). Both kernel CO and kernel CCO are sensitive to contaminated data, even
when bounded positive definite kernels are used. To the best of our knowledge,
there are few well-founded robust kernel methods for statistical unsupervised
learning. In addition, while the influence function (IF) of an estimator can
characterize its robustness, asymptotic properties and standard error, the IF
of a standard kernel canonical correlation analysis (standard kernel CCA) has
not been derived yet. To fill this gap, we first propose a robust kernel
covariance operator (robust kernel CO) and a robust kernel cross-covariance
operator (robust kernel CCO) based on a generalized loss function instead of
the quadratic loss function. Second, we derive the IF for robust kernel CCO and
standard kernel CCA. Using the IF of the standard kernel CCA, we can detect
influential observations from two sets of data. Finally, we propose a method
based on the robust kernel CO and the robust kernel CCO, called {\bf robust
kernel CCA}, which is less sensitive to noise than the standard kernel CCA. The
introduced principles can also be applied to many other kernel methods
involving kernel CO or kernel CCO. Our experiments on synthesized data and
imaging genetics analysis demonstrate that the proposed IF of standard kernel
CCA can identify outliers. It is also seen that the proposed robust kernel CCA
method performs better for ideal and contaminated data than the standard kernel
CCA.
