Complex Diffusion Maps with $ω$-Parameterized Kernels Revealing Inherent Harmonic Representations
For practitioners analyzing high-dimensional data (e.g., fMRI, EEG), CDM offers a novel method that amplifies differences among confusable samples and remains robust in high-noise settings, though the improvements are incremental over existing diffusion-based methods.
Complex Diffusion Maps (CDM) introduces a family of ω-parameterized complex-valued kernels that reveal dominant complex harmonics in high-dimensional data, improving discriminative power and robustness to noise. On a public EEG sleep dataset, CDM achieves competitive performance without task-specific tuning, while maintaining high computational efficiency.
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 Schrödinger kernel relevant to the Schrödinger equation, we propose a unified family of $ω$-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.