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

arXiv: https://arxiv.org/abs/2510.17561 · HTML: https://mlsanity.com/paper/2510.17561/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 8 of its 47 references that are in the map.
- First posted on arXiv 20 Oct 2025.

## Builds on (most cited first, 8 of 8)

- [Multimodal Deep Learning](https://mlsanity.com/paper/2301.04856/) (2023)
- [Optimality and Sub-optimality of PCA I: Spiked Random Matrix Models](https://mlsanity.com/paper/1807.00891/) (2018)
- [Optimal Inference in Contextual Stochastic Block Models](https://mlsanity.com/paper/2306.07948/) (2023)
- [Canonical Correlation Analysis (CCA) Based Multi-View Learning: An Overview](https://mlsanity.com/paper/1907.01693/) (2019)
- [Fundamental limits of community detection from multi-view data: multi-layer, dynamic and partially labeled block models](https://mlsanity.com/paper/2401.08167/) (2024)
- [Simultaneous Dimensionality Reduction: A Data Efficient Approach for Multimodal Representations Learning](https://mlsanity.com/paper/2310.04458/) (2023)
- [Computational Thresholds in Multi-Modal Learning via the Spiked Matrix-Tensor Model](https://mlsanity.com/paper/2506.02664/) (2025)
- [Optimal thresholds and algorithms for a model of multi-modal learning in high dimensions](https://mlsanity.com/paper/2407.03522/) (2024)

## Built on by (most cited first, 5 of 5)

- [Multi-layer Cross-attention is Provably Optimal for Multi-modal In-context Learning](https://mlsanity.com/paper/2602.04872/) (2026)
- [Information bottleneck for learning the phase space of dynamics from high-dimensional experimental data](https://mlsanity.com/paper/2604.24662/) (2026)
- [When to Align, When to Predict: A Phase Diagram for Multimodal Learning](https://mlsanity.com/paper/2606.11190/) (2026)
- [High-Dimensional Partial Least Squares: Spectral Analysis and Fundamental Limitations](https://mlsanity.com/paper/2512.15684/) (2025)
- [Achieving the Kesten-Stigum bound in the non-uniform hypergraph stochastic block model](https://mlsanity.com/paper/2604.20907/) (2026)

## Abstract (arXiv)

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-Peche (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.