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

arXiv: https://arxiv.org/abs/2605.01691 · HTML: https://mlsanity.com/paper/2605.01691/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 10 of its 70 references that are in the map.
- First posted on arXiv 3 May 2026.

## Builds on (most cited first, 10 of 10)

- [Semi-Supervised Classification with Graph Convolutional Networks](https://mlsanity.com/paper/1609.02907/) (2016)
- [Long Range Arena: A Benchmark for Efficient Transformers](https://mlsanity.com/paper/2011.04006/) (2020)
- [MagNet: A Neural Network for Directed Graphs](https://mlsanity.com/paper/2102.11391/) (2021)
- [Multi-View Spatial-Temporal Graph Convolutional Networks with Domain Generalization for Sleep Stage Classification](https://mlsanity.com/paper/2109.01824/) (2021)
- [SalientSleepNet: Multimodal Salient Wave Detection Network for Sleep Staging](https://mlsanity.com/paper/2105.13864/) (2021)
- [Magnetic Eigenmaps for the Visualization of Directed Networks](https://mlsanity.com/paper/1606.08266/) (2016)
- [Mixed Neural Network Approach for Temporal Sleep Stage Classification](https://mlsanity.com/paper/1610.06421/) (2016)
- [Data-driven Control of Agent-based Models: an Equation/Variable-free Machine Learning Approach](https://mlsanity.com/paper/2207.05779/) (2022)
- [Complex networks with complex weights](https://mlsanity.com/paper/2212.06257/) (2022)
- [Diffusion Representation for Asymmetric Kernels](https://mlsanity.com/paper/2401.12251/) (2024)

## Abstract (arXiv)

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 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.