Identifiability of Nonlinear Multi-view CCA under Partial Latent Observability: Limits and Guarantees of Recovery

We investigate the identifiability of nonlinear multi-view canonical correlation analysis (CCA) under partial latent observability. We formally show that existing pairwise identifiability does not guarantee simultaneous source recovery using a single encoder per view. When the representation dimension matches the source dimension, sum-of-correlations (SUMCOR), a generalized CCA objective variant, can favor higher-order modes over first-order source directions, causing representation drift. Under isotropic Gaussian prior and full cross-view latent coverage, we establish three recovery regimes at population global optima. First, global first-order dominance guarantees recovery of each view-specific source up to an affine transformation. Second, known overlap ranks enable Top-$r$ and masked SUMCOR to achieve the same recovery under pairwise first-order dominance. Third, even without either dominance condition, SUMCOR admits exact affine source readouts at finite, spectrum-dependent representation dimensions under a joint spectral compatibility condition. The enlarged representations retain source directions alongside stronger nonlinear modes. Our results demonstrate how spectral structure, rank information, and representation capacity govern source recovery. Synthetic experiments confirm the predicted recovery regimes and representation drift.