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 signal detection, establishing a sharp phase transition. Our central result proves that this threshold 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 significant risk of misinterpretation for scientists. Furthermore, we show that in the detectable phase, the second eigenvector aligns with the primary noise direction, revealing a deeper reorganisation of the system's structure. We complete our analysis with a Central Limit Theorem for the alignment fluctuations. 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