Provable Affine Identifiability of Nonlinear CCA under Latent Distributional Priors

Zhiwei Han, Stefan Matthes, Hao Shen
Proceedings of The 29th International Conference on Artificial Intelligence and Statistics, PMLR 300:3835-3843, 2026.

Abstract

In this work, we establish the sufficient conditions under which nonlinear Canonical Correlation Analysis (CCA) recovers ground-truth latent factors up to an affine transformation. By transporting the analysis from the observation space to the source space, we extend classical statistical results on orthogonal polynomial expansions of bivariate distributions to representation learning, proving affine identifiability under specific distributional priors. We formally demonstrate that whitening is strictly necessary to ensure the boundedness and well-conditioning of the learned mappings. Furthermore, we bridge the gap between theory and practice by proving that ridge-regularized empirical CCA converges to its population counterpart in the finite-sample regime. Finally, our findings provide a rigorous theoretical foundation explaining the empirical success of recent correlation-based non-contrastive learning methods. Experiments on synthetic and rendered image datasets, alongside systematic ablations, validate the predicted recovery behavior and illustrate the failure modes that arise when the assumptions are violated.

Cite this Paper

BibTeX
@InProceedings{pmlr-v300-han26b, title = { Provable Affine Identifiability of Nonlinear CCA under Latent Distributional Priors }, author = {Han, Zhiwei and Matthes, Stefan and Shen, Hao}, booktitle = {Proceedings of The 29th International Conference on Artificial Intelligence and Statistics}, pages = {3835--3843}, year = {2026}, editor = {Khan, Emtiyaz and Li, Yingzhen and Solin, Arno and Ramdas, Aaditya}, volume = {300}, series = {Proceedings of Machine Learning Research}, month = {02--05 May}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v300/main/assets/han26b/han26b.pdf}, url = {https://proceedings.mlr.press/v300/han26b.html}, abstract = { In this work, we establish the sufficient conditions under which nonlinear Canonical Correlation Analysis (CCA) recovers ground-truth latent factors up to an affine transformation. By transporting the analysis from the observation space to the source space, we extend classical statistical results on orthogonal polynomial expansions of bivariate distributions to representation learning, proving affine identifiability under specific distributional priors. We formally demonstrate that whitening is strictly necessary to ensure the boundedness and well-conditioning of the learned mappings. Furthermore, we bridge the gap between theory and practice by proving that ridge-regularized empirical CCA converges to its population counterpart in the finite-sample regime. Finally, our findings provide a rigorous theoretical foundation explaining the empirical success of recent correlation-based non-contrastive learning methods. Experiments on synthetic and rendered image datasets, alongside systematic ablations, validate the predicted recovery behavior and illustrate the failure modes that arise when the assumptions are violated. } }
Endnote
%0 Conference Paper %T Provable Affine Identifiability of Nonlinear CCA under Latent Distributional Priors %A Zhiwei Han %A Stefan Matthes %A Hao Shen %B Proceedings of The 29th International Conference on Artificial Intelligence and Statistics %C Proceedings of Machine Learning Research %D 2026 %E Emtiyaz Khan %E Yingzhen Li %E Arno Solin %E Aaditya Ramdas %F pmlr-v300-han26b %I PMLR %P 3835--3843 %U https://proceedings.mlr.press/v300/han26b.html %V 300 %X In this work, we establish the sufficient conditions under which nonlinear Canonical Correlation Analysis (CCA) recovers ground-truth latent factors up to an affine transformation. By transporting the analysis from the observation space to the source space, we extend classical statistical results on orthogonal polynomial expansions of bivariate distributions to representation learning, proving affine identifiability under specific distributional priors. We formally demonstrate that whitening is strictly necessary to ensure the boundedness and well-conditioning of the learned mappings. Furthermore, we bridge the gap between theory and practice by proving that ridge-regularized empirical CCA converges to its population counterpart in the finite-sample regime. Finally, our findings provide a rigorous theoretical foundation explaining the empirical success of recent correlation-based non-contrastive learning methods. Experiments on synthetic and rendered image datasets, alongside systematic ablations, validate the predicted recovery behavior and illustrate the failure modes that arise when the assumptions are violated.
APA
Han, Z., Matthes, S. & Shen, H.. (2026). Provable Affine Identifiability of Nonlinear CCA under Latent Distributional Priors . Proceedings of The 29th International Conference on Artificial Intelligence and Statistics, in Proceedings of Machine Learning Research 300:3835-3843 Available from https://proceedings.mlr.press/v300/han26b.html.

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