Spectral Thresholds in Correlated Spiked Models and Fundamental Limits of Partial Least Squares

Pierre Mergny, Lenka Zdeborová
Proceedings of The 29th International Conference on Artificial Intelligence and Statistics, PMLR 300:3925-3933, 2026.

Abstract

We provide a rigorous random matrix theory analysis of spiked cross-covariance models where the signals across two high-dimensional data channels are partially aligned. These models are motivated by multi-modal learning and form the standard generative setting underlying Partial Least Squares (PLS), a widely used yet theoretically underdeveloped method. We show that the leading singular values of the sample cross-covariance matrix undergo a Baik–Ben Arous–P{é}ch{é} (BBP)-type phase transition, and we characterize the precise thresholds for the emergence of informative components. Our results yield the first sharp asymptotic description of the signal recovery capabilities of PLS in this setting, revealing a fundamental performance gap between PLS and the Bayes-optimal estimator. In particular, we identify the SNR and correlation regimes where PLS fails to recover any signal, despite detectability being possible in principle. These findings clarify the theoretical limits of PLS and provide guidance for the design of reliable multi-modal inference methods in high dimensions.

Cite this Paper

BibTeX
@InProceedings{pmlr-v300-mergny26a, title = { Spectral Thresholds in Correlated Spiked Models and Fundamental Limits of Partial Least Squares }, author = {Mergny, Pierre and Zdeborov{\'a}, Lenka}, booktitle = {Proceedings of The 29th International Conference on Artificial Intelligence and Statistics}, pages = {3925--3933}, 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/mergny26a/mergny26a.pdf}, url = {https://proceedings.mlr.press/v300/mergny26a.html}, abstract = { We provide a rigorous random matrix theory analysis of spiked cross-covariance models where the signals across two high-dimensional data channels are partially aligned. These models are motivated by multi-modal learning and form the standard generative setting underlying Partial Least Squares (PLS), a widely used yet theoretically underdeveloped method. We show that the leading singular values of the sample cross-covariance matrix undergo a Baik–Ben Arous–P{é}ch{é} (BBP)-type phase transition, and we characterize the precise thresholds for the emergence of informative components. Our results yield the first sharp asymptotic description of the signal recovery capabilities of PLS in this setting, revealing a fundamental performance gap between PLS and the Bayes-optimal estimator. In particular, we identify the SNR and correlation regimes where PLS fails to recover any signal, despite detectability being possible in principle. These findings clarify the theoretical limits of PLS and provide guidance for the design of reliable multi-modal inference methods in high dimensions. } }
Endnote
%0 Conference Paper %T Spectral Thresholds in Correlated Spiked Models and Fundamental Limits of Partial Least Squares %A Pierre Mergny %A Lenka Zdeborová %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-mergny26a %I PMLR %P 3925--3933 %U https://proceedings.mlr.press/v300/mergny26a.html %V 300 %X We provide a rigorous random matrix theory analysis of spiked cross-covariance models where the signals across two high-dimensional data channels are partially aligned. These models are motivated by multi-modal learning and form the standard generative setting underlying Partial Least Squares (PLS), a widely used yet theoretically underdeveloped method. We show that the leading singular values of the sample cross-covariance matrix undergo a Baik–Ben Arous–P{é}ch{é} (BBP)-type phase transition, and we characterize the precise thresholds for the emergence of informative components. Our results yield the first sharp asymptotic description of the signal recovery capabilities of PLS in this setting, revealing a fundamental performance gap between PLS and the Bayes-optimal estimator. In particular, we identify the SNR and correlation regimes where PLS fails to recover any signal, despite detectability being possible in principle. These findings clarify the theoretical limits of PLS and provide guidance for the design of reliable multi-modal inference methods in high dimensions.
APA
Mergny, P. & Zdeborová, L.. (2026). Spectral Thresholds in Correlated Spiked Models and Fundamental Limits of Partial Least Squares . Proceedings of The 29th International Conference on Artificial Intelligence and Statistics, in Proceedings of Machine Learning Research 300:3925-3933 Available from https://proceedings.mlr.press/v300/mergny26a.html.

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