Eigenvector Phase Transitions under Anisotropic Noise

TMLR Paper5609 Authors

Abstract: Identifying latent structures in environmental data—such as habitat clusters or pollution sources—is a fundamental challenge in ecological and climate science. Spectral methods, which analyse the principal eigenvectors of affinity matrices, are powerful tools for this task. However, environmental systems are rarely isotropic; physical processes like river flows or prevailing winds create strong directional gradients, resulting in anisotropic noise. The effect of such anisotropy on the reliability of spectral methods is not yet well understood in the literature. In this work, we develop a rigorous theory for this scenario by analysing a spiked random matrix model subjected to anisotropic noise. We derive an exact, analytical expression for the critical signal-to-noise ratio required for strong signal detection, establishing a sharp phase transition. We prove that this threshold is information-theoretically optimal, and that it depends critically on the geometric alignment between the signal and the dominant environmental gradient, formalising a camouflage effect''. We also uncover a critical failure mode where this environmental gradient can itself create a phantom'' structure that spectral methods can easily detect, posing a potential risk of misinterpretation for scientists. Furthermore, we show that in the detectable phase, the eigenspace undergoes a systematic reorganisation: the principal eigenvector aligns with the signal while the second eigenvector aligns with the primary noise direction. We complete our analysis with Central Limit Theorems for the alignment fluctuations of both the signal and noise eigenvectors. Finally, we propose and analyse a correction framework based on second-moment information, demonstrating a theoretical pathway to overcome the camouflage-induced bias and rigorously characterising its practical sensitivities. We validate our theoretical predictions with simulations of ecological systems, offering a fundamental understanding of when spectral methods succeed or fail in realistic environments. Code to reproduce all results in the paper is anonymously released at https://anonymous.4open.science/r/tmlr_ept

Submission Length: Long submission (more than 12 pages of main content)

Assigned Action Editor: ~Marco_Mondelli1

Submission Number: 5609