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

2025-10-20

AI Summary

This paper investigates signal recovery in cross-covariance modeling of high-dimensional partially aligned multimodal data, focusing on the fundamental performance limits of Partial Least Squares (PLS) under finite signal-to-noise ratio (SNR) and inter-modal correlation. Leveraging random matrix theory and the spiked model, we derive the first exact BBP-type phase transition threshold for the singular values of the sample cross-covariance matrix, identifying a critical regime where informative components are statistically detectable yet PLS fails. We provide the first precise asymptotic characterization of PLS’s signal recovery capability, revealing its intrinsic gap relative to the Bayesian optimal estimator. Furthermore, we systematically delineate the SNR–correlation parameter region in which PLS breaks down. These results establish a key theoretical benchmark for designing and evaluating high-dimensional multimodal inference methods.

Technology Category

• Machine Learning: Multimodal Learning
• Computer Vision: Multi-modal Vision
• Reasoning under Uncertainty: Relational Probabilistic Models

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Mining multimedia, multimodal, multilingual, cross-lingual Web data
• User Modeling, Personalization and Recommendation: User privacy protection in personalized systems

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-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.

Problem

• Analyzes signal recovery limits in high-dimensional cross-covariance models
• Characterizes phase transitions in Partial Least Squares performance
• Identifies SNR regimes where PLS fails despite detectable signals

Innovation

• Random matrix theory analyzes spiked cross-covariance models
• Characterizes phase transition thresholds for informative components
• Identifies SNR regimes where PLS fails signal recovery