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Nonlinear Sparse Generalized Canonical Correlation Analysis for Multi-view High-dimensional Data

Rong Wu, Ziqi Chen, Gen Li, Hai Shu

TL;DR

This work tackles the integration of multi-view high-dimensional biomedical data by developing nonlinear, sparse GCCA methods that extend two-view nonlinear approaches to K-view settings. The core method, HSIC-SGCCA, enforces a unit-variance constraint and solves a nonconvex optimization via a fusion of Block Prox-Linear and LADMM, while SA-KGCCA and TS-KGCCA provide multi-convex, kernelized alternatives solved by block-coordinate strategies. Empirical results on simulations and TCGA-BRCA demonstrate that HSIC-SGCCA achieves superior variable selection and enhances downstream tasks such as cancer subtype separation and survival prediction. The approaches offer practical tools for robust multi-view integration, with potential extensions into supervised and structure-informed variants to further improve interpretability and predictive performance.

Abstract

Motivation: Biomedical studies increasingly produce multi-view high-dimensional datasets (e.g., multi-omics) that demand integrative analysis. Existing canonical correlation analysis (CCA) and generalized CCA methods address at most two of the following three key aspects simultaneously: (i) nonlinear dependence, (ii) sparsity for variable selection, and (iii) generalization to more than two data views. There is a pressing need for CCA methods that integrate all three aspects to effectively analyze multi-view high-dimensional data. Results: We propose three nonlinear, sparse, generalized CCA methods, HSIC-SGCCA, SA-KGCCA, and TS-KGCCA, for variable selection in multi-view high-dimensional data. These methods extend existing SCCA-HSIC, SA-KCCA, and TS-KCCA from two-view to multi-view settings. While SA-KGCCA and TS-KGCCA yield multi-convex optimization problems solved via block coordinate descent, HSIC-SGCCA introduces a necessary unit-variance constraint previously ignored in SCCA-HSIC, resulting in a nonconvex, non-multiconvex problem. We efficiently address this challenge by integrating the block prox-linear method with the linearized alternating direction method of multipliers. Simulations and TCGA-BRCA data analysis demonstrate that HSIC-SGCCA outperforms competing methods in multi-view variable selection.

Nonlinear Sparse Generalized Canonical Correlation Analysis for Multi-view High-dimensional Data

TL;DR

This work tackles the integration of multi-view high-dimensional biomedical data by developing nonlinear, sparse GCCA methods that extend two-view nonlinear approaches to K-view settings. The core method, HSIC-SGCCA, enforces a unit-variance constraint and solves a nonconvex optimization via a fusion of Block Prox-Linear and LADMM, while SA-KGCCA and TS-KGCCA provide multi-convex, kernelized alternatives solved by block-coordinate strategies. Empirical results on simulations and TCGA-BRCA demonstrate that HSIC-SGCCA achieves superior variable selection and enhances downstream tasks such as cancer subtype separation and survival prediction. The approaches offer practical tools for robust multi-view integration, with potential extensions into supervised and structure-informed variants to further improve interpretability and predictive performance.

Abstract

Motivation: Biomedical studies increasingly produce multi-view high-dimensional datasets (e.g., multi-omics) that demand integrative analysis. Existing canonical correlation analysis (CCA) and generalized CCA methods address at most two of the following three key aspects simultaneously: (i) nonlinear dependence, (ii) sparsity for variable selection, and (iii) generalization to more than two data views. There is a pressing need for CCA methods that integrate all three aspects to effectively analyze multi-view high-dimensional data. Results: We propose three nonlinear, sparse, generalized CCA methods, HSIC-SGCCA, SA-KGCCA, and TS-KGCCA, for variable selection in multi-view high-dimensional data. These methods extend existing SCCA-HSIC, SA-KCCA, and TS-KCCA from two-view to multi-view settings. While SA-KGCCA and TS-KGCCA yield multi-convex optimization problems solved via block coordinate descent, HSIC-SGCCA introduces a necessary unit-variance constraint previously ignored in SCCA-HSIC, resulting in a nonconvex, non-multiconvex problem. We efficiently address this challenge by integrating the block prox-linear method with the linearized alternating direction method of multipliers. Simulations and TCGA-BRCA data analysis demonstrate that HSIC-SGCCA outperforms competing methods in multi-view variable selection.

Paper Structure

This paper contains 29 sections, 35 equations, 2 figures, 4 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. RKHS
  5. HSIC
  6. (G)CCA and S(G)CCA
  7. HSIC-based (S)CCA
  8. Methods
  9. HSIC-SGCCA Problem Formulation
  10. HSIC-SGCCA Algorithm Development
  11. Simulations
  12. Simulation settings
  13. Evaluation metrics
  14. Simulation results
  15. Applications to TCGA-BRCA data
  16. ...and 14 more sections

Figures (2)

  • Figure 1: Simulation results (Average $\pm$ 1.96SE) based on 100 independent replications. Small error bars may be hidden by the average dots. Runtime results of SA-KGCCA are provided in Supplementary Table S1 due to large values.
  • Figure 2: Venn diagrams showing the numbers of TCGA-BRCA variables selected by the six methods. Total counts are provided in Table \ref{['tabvs']}.