D-CCA: A Decomposition-Based Canonical Correlation Analysis for High-Dimensional Datasets

ABSTRACT

A typical approach to the joint analysis of two high-dimensional datasets is to decompose each data matrix into three parts: a low-rank common matrix that captures the shared information across datasets, a low-rank distinctive matrix that characterizes the individual information within a single dataset, and an additive noise matrix. Existing decomposition methods often focus on the orthogonality between the common and distinctive matrices, but inadequately consider the more necessary orthogonal relationship between the two distinctive matrices. The latter guarantees that no more shared information is extractable from the distinctive matrices. We propose decomposition-based canonical correlation analysis (D-CCA), a novel decomposition method that defines the common and distinctive matrices from the ${\ell }^2$ space of random variables rather than the conventionally used Euclidean space, with a careful construction of the orthogonal relationship between distinctive matrices. D-CCA represents a natural generalization of the traditional canonical correlation analysis. The proposed estimators of common and distinctive matrices are shown to be consistent and have reasonably better performance than some state-of-the-art methods in both simulated data and the real data analysis of breast cancer data obtained from The Cancer Genome Atlas. Supplementary materials for this article are available online.

KEYWORDS:

• Approximate factor model
• Canonical variable
• Common structure
• Distinctive structure
• Soft thresholding

Additional information

Funding

Dr. Zhu’s work was partially supported by NIH grants MH086633 and MH116527, NSF grants SES-1357666 and DMS-1407655, a grant from the Cancer Prevention Research Institute of Texas, and the endowed Bao-Shan Jing Professorship in Diagnostic Imaging. The content is solely the responsibility of the authors and does not necessarily represent the official views of the NIH or any other funding agency.