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

TL;DR

This paper introduces Complex Diffusion Maps (CDM), a new framework using complex-valued kernels to reveal dominant harmonic structures in high-dimensional data, improving discrimination and robustness.

Contribution

The paper develops a unified family of $ ext{omega}$-parameterized complex kernels for diffusion mapping, with a theoretical foundation and optimization interpretation, enhancing data analysis capabilities.

Findings

CDM amplifies differences among confusable samples, improving discriminative power.

CDM is robust in high-noise environments, with clearer spectral separation.

CDM captures nonlocal spatiotemporal dynamics in resting-state fMRI data.

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.