Complex Diffusion Maps with $\omega$-Parameterized Kernels Revealing Inherent Harmonic Representations

arXiv cs.LG / 5/5/2026

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Key Points

The paper introduces Complex Diffusion Maps (CDM), a diffusion-mapping framework designed to uncover dominant complex harmonic structure in high-dimensional data.
CDM uses a unified family of ω-parameterized complex-valued kernels that balance local (heat-equation–like) and nonlocal (Schrödinger-equation–like) relationships, with a rigorous operator-spectrum-theory foundation.
The authors define diffusion operators, diffusion distances, and complex harmonic maps in a mathematically well-posed way, and provide an optimization-based interpretation focused on preserving angular structure in the complex diffusion space.
Experiments on synthetic and real datasets show improved discriminative power versus methods using real-valued kernels, robustness under high noise via clearer eigengaps, and stronger recovery of nonlocal spatiotemporal dynamics in resting-state fMRI.
For EEG sleep data, CDM performs competitively without task-specific tuning while remaining computationally efficient relative to many traditional ML and deep neural network approaches, supporting its practicality and generality.

Abstract

In this paper, we propose Complex Diffusion Maps (CDM), a novel diffusion mapping framework that aims to reveal the dominant complex harmonics of high-dimensional data. Inspired by the local Gaussian kernel relevant to the heat equation and the nonlocal Schr\"odinger kernel relevant to the Schr\"odinger equation, we propose a unified family of

\omega

-parameterized complex-valued kernels for the trade-off between local and nonlocal connections. We establish the theoretical foundation based on the operator spectrum theory, where the corresponding diffusion operator, diffusion distance, and complex harmonic maps are well-defined. An optimization-based interpretation of the maps is also developed, aiming to preserve angular structure in the complex diffusion space rather than relying solely on real-valued magnitude. We extensively evaluate CDM on both synthetic and real-world datasets. The complex-valued kernel amplifies differences among easily confusable samples, improving discriminative power over both linear and nonlinear methods based on real-valued kernels. CDM remains robust in high-noise settings, yielding a clearer eigengap that enhances spectral separation. For resting-state fMRI data, CDM captures more strongly correlated and nonlocal spatiotemporal dynamics. Without task-specific tuning, CDM achieves competitive performance on a public EEG sleep dataset, while maintaining high computational efficiency compared with both traditional machine learning and deep neural network approaches, highlighting its generality and practical value.