Provable Subspace Identification of Nonlinear Multi-view CCA

2026-02-27

AI Summary

This work addresses the identifiability challenge in nonlinear multi-view canonical correlation analysis (CCA), where nonlinear mixing renders the shared latent subspace unidentifiable. The problem is reformulated as a basis-invariant subspace identification task. Under mild assumptions on the latent prior and spectral separability, the proposed method uniquely recovers the globally shared signal subspace in settings with three or more views while effectively suppressing view-specific noise. Theoretically, this study establishes the first identifiability guarantees for the shared subspace in nonlinear multi-view settings, deriving explicit subspace error bounds and finite-sample consistency by integrating spectral perturbation theory, concentration properties of cross-covariance operators, and modeling of orthogonal ambiguities. Experiments on both synthetic and rendered image data validate the method’s efficacy and confirm the necessity of the key theoretical assumptions.

Technology Category

• Machine Learning: Multi-instance/Multi-view Learning
• Computer Vision: Multi-modal Vision
• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Security and Privacy: Large-scale security measurements
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

We investigate the identifiability of nonlinear Canonical Correlation Analysis (CCA) in a multi-view setup, where each view is generated by an unknown nonlinear map applied to a linear mixture of shared latents and view-private noise. Rather than attempting exact unmixing, a problem proven to be ill-posed, we instead reframe multi-view CCA as a basis-invariant subspace identification problem. We prove that, under suitable latent priors and spectral separation conditions, multi-view CCA recovers the pairwise correlated signal subspaces up to view-wise orthogonal ambiguity. For $N \geq 3$ views, the objective provably isolates the jointly correlated subspaces shared across all views while eliminating view-private variations. We further establish finite-sample consistency guarantees by translating the concentration of empirical cross-covariances into explicit subspace error bounds via spectral perturbation theory. Experiments on synthetic and rendered image datasets validate our theoretical findings and confirm the necessity of the assumed conditions.

Problem

• subspace identification
• nonlinear CCA
• multi-view learning
• latent variables
• identifiability

Innovation

• subspace identification
• nonlinear multi-view CCA
• identifiability
• spectral perturbation theory
• finite-sample consistency