


























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 $\omega$-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.
| Comments: | 27 pages main text, 13 pages appendix, 9 figures, 2 tables. Submitted to IEEE TPAMI. Code will be made publicly available upon acceptance |
| Subjects: | Machine Learning (cs.LG) |
| MSC classes: | 68T10, 68R12, 62H30 |
| Cite as: | arXiv:2605.01691 [cs.LG] |
| (or arXiv:2605.01691v1 [cs.LG] for this version) | |
| https://doi.org/10.48550/arXiv.2605.01691 arXiv-issued DOI via DataCite (pending registration) |
From: Weiyang Ding [view email]
[v1]
Sun, 3 May 2026 03:08:24 UTC (7,691 KB)
此内容由惯性聚合(RSS阅读器)自动聚合整理,仅供阅读参考。 原文来自 — 版权归原作者所有。