Provable Subspace Identification of Nonlinear Multi-view CCA

TL;DR

This paper provides a theoretical framework for identifying shared subspaces in nonlinear multi-view CCA, demonstrating conditions under which the method reliably recovers correlated signal structures.

Contribution

It introduces a provable subspace identification approach for nonlinear multi-view CCA, establishing conditions for recoverability and finite-sample guarantees.

Findings

Recoverability of shared subspaces under spectral separation conditions

Finite-sample consistency guarantees with explicit error bounds

Validation on synthetic and image datasets confirms theoretical results

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.