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Provable Affine Identifiability of Nonlinear CCA under Latent Distributional Priors

Zhiwei Han, Stefan Matthes, Hao Shen

TL;DR

The paper proves that nonlinear CCA can identifiably recover ground-truth latent factors up to an affine transform when latent priors are present and whitening is applied. It establishes population affine identifiability for Gaussian priors and shows ridge-regularized empirical CCA converges to the population solution as data grows. The analysis hinges on transporting the problem to source space and exploiting spectral separation via Hermite/Mehler expansions, with reparameterization invariance guaranteeing consistency across representations. Empirical results on synthetic and rendered data validate the theory and demonstrate the importance of whitening, while ablations reveal the limits under assumption violations. Overall, the work positions affine identifiability as a principled alternative to contrastive methods for disentangling multi-view latent structure in nonlinear settings.

Abstract

In this work, we establish conditions under which nonlinear CCA recovers the ground-truth latent factors up to an orthogonal transform after whitening. Building on the classical result that linear mappings maximize canonical correlations under Gaussian priors, we prove affine identifiability for a broad class of latent distributions in the population setting. Central to our proof is a reparameterization result that transports the analysis from observation space to source space, where identifiability becomes tractable. We further show that whitening is essential for ensuring boundedness and well-conditioning, thereby underpinning identifiability. Beyond the population setting, we prove that ridge-regularized empirical CCA converges to its population counterpart, transferring these guarantees to the finite-sample regime. Experiments on a controlled synthetic dataset and a rendered image dataset validate our theory and demonstrate the necessity of its assumptions through systematic ablations.

Provable Affine Identifiability of Nonlinear CCA under Latent Distributional Priors

TL;DR

The paper proves that nonlinear CCA can identifiably recover ground-truth latent factors up to an affine transform when latent priors are present and whitening is applied. It establishes population affine identifiability for Gaussian priors and shows ridge-regularized empirical CCA converges to the population solution as data grows. The analysis hinges on transporting the problem to source space and exploiting spectral separation via Hermite/Mehler expansions, with reparameterization invariance guaranteeing consistency across representations. Empirical results on synthetic and rendered data validate the theory and demonstrate the importance of whitening, while ablations reveal the limits under assumption violations. Overall, the work positions affine identifiability as a principled alternative to contrastive methods for disentangling multi-view latent structure in nonlinear settings.

Abstract

In this work, we establish conditions under which nonlinear CCA recovers the ground-truth latent factors up to an orthogonal transform after whitening. Building on the classical result that linear mappings maximize canonical correlations under Gaussian priors, we prove affine identifiability for a broad class of latent distributions in the population setting. Central to our proof is a reparameterization result that transports the analysis from observation space to source space, where identifiability becomes tractable. We further show that whitening is essential for ensuring boundedness and well-conditioning, thereby underpinning identifiability. Beyond the population setting, we prove that ridge-regularized empirical CCA converges to its population counterpart, transferring these guarantees to the finite-sample regime. Experiments on a controlled synthetic dataset and a rendered image dataset validate our theory and demonstrate the necessity of its assumptions through systematic ablations.

Paper Structure

This paper contains 24 sections, 3 theorems, 10 equations, 6 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Related Work
  3. Disentangled Representation Learning.
  4. Nonlinear CCA.
  5. Canonical Correlation Analysis for Disentangled Representation
  6. Data Generating Process
  7. Canonical Correlation Analysis
  8. CCA Population Objective.
  9. Population Affine Identifiability
  10. Empirical Estimation and Statistical Consistency
  11. Experiments
  12. Experimental Setup
  13. Datasets.
  14. Baselines.
  15. Metrics.
  16. ...and 9 more sections

Key Result

Proposition 1

Let $\mathcal{S},\mathcal{X},\mathcal{X}'$ be standard Borel spaces and $\mathbf{g}, \mathbf{g}'$ be Borel-measurable and injective. In addition, let $\tilde{\mathcal{F}}_{\mathcal{X}},\tilde{\mathcal{F}}'_{\mathcal{X}'}$ be the whitened encoder classes of Assumption assump:function-class and assume and the feasible latent whitened classes, for $\mathbf u\in\{\mathbf s,\mathbf s'\}$, define Furth

Figures (6)

  • Figure 1: Whitened latent learned by DeepCCA on synthetic data ($d_\mathcal{S}=d_\mathcal{Z}=2$). Color gradients in different rows illustrate the variations along a single coordinate in $\mathcal{S}$.
  • Figure 2: The $\ell_\infty$-gap in singular values and log orbit distance over training steps in Gaussian case. Shaded regions denote $\pm1$ standard deviation across runs.
  • Figure 3: $R^2\uparrow$ and Orbit distance over changing sample size in fixed dataset setup.
  • Figure 4: Ablation over the source dimension $d_\mathcal{S}$ ($d_\mathcal{S} =d_\mathcal{Z}$). Left: $R^2\uparrow$. Solid lines denote encoder $\mathbf{f}$ and dashed lines encoder $\mathbf{f}'$. Right: log principal angles of $\mathbf{f}$ in the Gaussian case. Black dots denote principal angles and shaded region indicates the log-standard deviation.
  • Figure 5: Ablation over the first order canonical ratio $\rho_{d_{\mathcal{S}}}/\rho_1^2$ ($d_\mathcal{S} =d_\mathcal{Z}$). Left: $R^2\uparrow$. Right: $PA_{max}\downarrow$. Colors denote different source dimensions.
  • ...and 1 more figures

Theorems & Definitions (3)

  • Proposition 1: Reparameterization Invariance and Representational Universality of CCA
  • Theorem 1: Population Affine Identifiability
  • Theorem 2: Consistency of Empirical Maximizers