  Integrative analysis of multiple heterogeneous datasets has become standard
practice in many research fields, especially in single-cell genomics and
medical informatics. Existing approaches oftentimes suffer from limited power
in capturing nonlinear structures, insufficient account of noisiness and
effects of high-dimensionality, lack of adaptivity to signals and sample sizes
imbalance, and their results are sometimes difficult to interpret. To address
these limitations, we propose a novel kernel spectral method that achieves
joint embeddings of two independently observed high-dimensional noisy datasets.
The proposed method automatically captures and leverages possibly shared
low-dimensional structures across datasets to enhance embedding quality. The
obtained low-dimensional embeddings can be utilized for many downstream tasks
such as simultaneous clustering, data visualization, and denoising. The
proposed method is justified by rigorous theoretical analysis. Specifically, we
show the consistency of our method in recovering the low-dimensional noiseless
signals, and characterize the effects of the signal-to-noise ratios on the
rates of convergence. Under a joint manifolds model framework, we establish the
convergence of ultimate embeddings to the eigenfunctions of some newly
introduced integral operators. These operators, referred to as duo-landmark
integral operators, are defined by the convolutional kernel maps of some
reproducing kernel Hilbert spaces (RKHSs). These RKHSs capture the either
partially or entirely shared underlying low-dimensional nonlinear signal
structures of the two datasets. Our numerical experiments and analyses of two
single-cell omics datasets demonstrate the empirical advantages of the proposed
method over existing methods in both embeddings and several downstream tasks.
