RKUM is an R package developed for implementing robust kernel-based unsupervised methods. It provides functions for estimating the robust kernel covariance operator (CO) and the robust kernel cross-covariance operator (CCO) using generalized loss functions instead of the conventional quadratic loss. These operators form the foundation of robust kernel learning and enable reliable analysis under contaminated or noisy data conditions. The package includes implementations of robust kernel canonical correlation analysis (Kernel CCA), as well as the influence function (IF) for both standard and multiple kernel CCA frameworks. The influence function quantifies sensitivity and helps detect influential or outlying observations across two-view and multi-view datasets. Experiments using synthesized two-view and multi-view data demonstrate that the IF of the standard kernel CCA effectively identifies outliers, while the robust kernel methods implemented in RKUM exhibit reduced sensitivity to contamination. Overall, RKUM provides an efficient and extensible platform for robust kernel-based analysis in high-dimensional data applications.

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.

In genome-wide interaction studies, to detect gene-gene interactions, most methods are divided into two folds: single nucleotide polymorphisms (SNP) based and gene-based methods. Basically, the methods based on the gene are more effective than the methods based on a single SNP. Recent years, while the kernel canonical correlation analysis (Classical kernel CCA) based U statistic (KCCU) has proposed to detect the nonlinear relationship between genes. To estimate the variance in KCCU, they have used resampling based methods which are highly computationally intensive. In addition, classical kernel CCA is not robust to contaminated data. We, therefore, first discuss robust kernel mean element, the robust kernel covariance, and cross-covariance operators. Second, we propose a method based on influence function to estimate the variance of the KCCU. Third, we propose a nonparametric robust KCCU method based on robust kernel CCA, which is designed for contaminated data and less sensitive to noise than classical kernel CCA. Finally, we investigate the proposed methods to synthesized data and imaging genetic data set. Based on gene ontology and pathway analysis, the synthesized and genetics analysis demonstrate that the proposed robust method shows the superior performance of the state-of-the-art methods.

Imaging genetic research has essentially focused on discovering unique and co-association effects, but typically ignoring to identify outliers or atypical objects in genetic as well as non-genetics variables. Identifying significant outliers is an essential and challenging issue for imaging genetics and multiple sources data analysis. Therefore, we need to examine for transcription errors of identified outliers. First, we address the influence function (IF) of kernel mean element, kernel covariance operator, kernel cross-covariance operator, kernel canonical correlation analysis (kernel CCA) and multiple kernel CCA. Second, we propose an IF of multiple kernel CCA, which can be applied for more than two datasets. Third, we propose a visualization method to detect influential observations of multiple sources of data based on the IF of kernel CCA and multiple kernel CCA. Finally, the proposed methods are capable of analyzing outliers of subjects usually found in biomedical applications, in which the number of dimension is large. To examine the outliers, we use the stem-and-leaf display. Experiments on both synthesized and imaging genetics data (e.g., SNP, fMRI, and DNA methylation) demonstrate that the proposed visualization can be applied effectively.

To the best of our knowledge, there are no general well-founded robust methods for statistical unsupervised learning. Most of the unsupervised methods explicitly or implicitly depend on the kernel covariance operator (kernel CO) or kernel cross-covariance operator (kernel CCO). They are sensitive to contaminated data, even when using bounded positive definite kernels. First, we propose robust kernel covariance operator (robust kernel CO) and robust kernel crosscovariance operator (robust kernel CCO) based on a generalized loss function instead of the quadratic loss function. Second, we propose influence function of classical kernel canonical correlation analysis (classical kernel CCA). Third, using this influence function, we propose a visualization method to detect influential observations from two sets of data. Finally, we propose a method based on robust kernel CO and robust kernel CCO, called robust kernel CCA, which is designed for contaminated data and less sensitive to noise than classical kernel CCA. The principles we describe also apply to many kernel methods which must deal with the issue of kernel CO or kernel CCO. Experiments on synthesized and imaging genetics analysis demonstrate that the proposed visualization and robust kernel CCA can be applied effectively to both ideal data and contaminated data. The robust methods show the superior performance over the state-of-the-art methods.