Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Provable Subspace Identification of Nonlinear Multi-view CCA

Zhiwei Han, Stefan Matthes, Hao Shen

TL;DR

It is proved that, under suitable latent priors and spectral separation conditions, multi-view CCA recovers the pairwise correlated signal subspaces up to view-wise orthogonal ambiguity.

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.

Provable Subspace Identification of Nonlinear Multi-view CCA

TL;DR

It is proved that, under suitable latent priors and spectral separation conditions, multi-view CCA recovers the pairwise correlated signal subspaces up to view-wise orthogonal ambiguity.

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 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.
Paper Structure (68 sections, 13 theorems, 180 equations, 2 figures, 7 tables)

This paper contains 68 sections, 13 theorems, 180 equations, 2 figures, 7 tables.

Table of Contents

  1. Introduction
  2. Related Work
  3. Identifiable Disentanglement and Nonlinear ICA.
  4. Identifiability from Multiple Views.
  5. Identifiability of Nonlinear Multi-view CCA.
  6. Problem Formulation
  7. Multi-view Generative Process.
  8. Latent Structure.
  9. Learning Objective.
  10. Notations.
  11. Preliminaries
  12. Nonlinear Multi-view CCA
  13. Generalized CCA for Multi-view Learning.
  14. CCA in the source domain.
  15. Whitening and Canonicalization
  16. ...and 53 more sections

Key Result

Lemma 1

Under the isotropic Gaussian prior in ass:isotropy, define the canonical coordinates and denote $(\mathbf U_{ij}, \mathbf V_{ij})$ as the corresponding canonicalizers, By the change-of-variables formula, the induced joint density is Moreover, $p_{\mathbf u,\mathbf v}$ factorizes as where $\phi$ is the standard normal density and $\phi_t$ is the standardized bivariate normal density with correla

Figures (2)

  • Figure 1: Ablation over the first-order canonical ratio $\rho_{d_{\mathcal{S}}}/\rho_1^2$ ($d_{\mathcal{S}}=d_{\mathcal{Z}}$). Left and right panels display the mean ($PA_{\mathrm{mean}}\downarrow$) and maximum ($PA_{\mathrm{max}}\downarrow$) principal angles, respectively. Colors indicate source dimensions, while solid, dashed, and dash-dotted lines denote the view-specific encoders $\tilde{\mathbf{f}}_1$, $\tilde{\mathbf{f}}_2$, and $\tilde{\mathbf{f}}_3$.
  • Figure 2: Log principal angles of encoder $\mathbf{f}_1$ in the under-complete setup ($d_{\mathcal{S}_i}=5,\ d_{\mathcal{Z}}=4, \forall i \in [3]$). Black dots denote principal angles and the shaded region indicates the log-standard deviation.

Theorems & Definitions (31)

  • Lemma 1: Canonical Factorization
  • Remark 1: Information-Theoretic Interpretation
  • Lemma 2: Normalized Multivariate Mehler-Hermite Expansion
  • Definition 1: Signal and Correlated Subspaces
  • Theorem 5.1: Infinite-Dimensional Two-view Subspace Identifiability
  • Corollary 1: Finite-Dimensional Two-view Subspace Identifiability
  • Remark 2: Spectral separation of higher-order components
  • Theorem 5.2: Infinite-Dimensional Multi-View Subspace Identifiability
  • Remark 3: Subspace dimensionality and partial observability
  • Theorem 5.3: Finite-sample subspace recovery
  • ...and 21 more