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

• URL: http://arxiv.org/abs/2605.01691v1
• Date: Sun, 03 May 2026 03:08:24 GMT
• Title: Complex Diffusion Maps with $ω$-Parameterized Kernels Revealing Inherent Harmonic Representations
• Abstract summary: Complex Diffusion Maps (CDM) is a novel diffusion mapping framework that aims to reveal the dominant complex harmonics of high-dimensional data.<n>We extensively evaluate CDM on both synthetic and real-world datasets.
• Score: 17.041810980453246
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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 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.

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