Influence Function and Robust Variant of Kernel Canonical Correlation Analysis
This work addresses robustness issues in kernel methods for unsupervised learning, which is important for applications like imaging genetics, but it is incremental as it builds on existing kernel CCA frameworks.
The paper tackles the sensitivity of kernel covariance and cross-covariance operators to contaminated data in unsupervised learning by proposing robust versions and deriving influence functions for kernel canonical correlation analysis (CCA). The results show that the robust kernel CCA method outperforms standard kernel CCA on both ideal and contaminated data, and the influence function can effectively identify outliers.
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.