Kernel Spectral Joint Embeddings for High-Dimensional Noisy Datasets Using Duo-Landmark Integral Operators

Integrative analysis of multiple heterogeneous datasets has arised in many research fields. 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 kernel spectral method that achieves joint embeddings of two independently observed high-dimensional noisy datasets. The proposed method automatically captures and leverages shared low-dimensional structures across datasets to enhance embedding quality. The obtained low-dimensional embeddings can be used for downstream tasks such as simultaneous clustering, data visualization, and denoising. The proposed method is justified by rigorous theoretical analysis, which guarantees its consistency in capturing the signal structures, and provides a geometric interpretation of the embeddings. Under a joint manifolds model framework, we establish the convergence of the embeddings to the eigenfunctions of some natural 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 underlying, shared low-dimensional nonlinear signal structures between the two datasets. Our numerical experiments and analyses of two pairs of single-cell omics datasets demonstrate the empirical advantages of the proposed method over existing methods in both embeddings and several downstream tasks. Supplementary materials for this article are available online, including a standardized description of the materials available for reproducing the work.