# Reconstruction of Graph Signals on Complex Manifolds with Kernel Methods

arXiv: https://arxiv.org/abs/2505.15202 · HTML: https://mlsanity.com/paper/2505.15202/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 18 of its 57 references that are in the map.
- First posted on arXiv 21 May 2025.

## Builds on (most cited first, 18 of 18)

- [Graph Signal Processing: Overview, Challenges and Applications](https://mlsanity.com/paper/1712.00468/) (2017)
- [Discrete Signal Processing on Graphs: Frequency Analysis](https://mlsanity.com/paper/1307.0468/) (2013)
- [Graph Signal Processing: History, Development, Impact, and Outlook](https://mlsanity.com/paper/2303.12211/) (2023)
- [Discrete Signal Processing on Graphs: Sampling Theory](https://mlsanity.com/paper/1503.05432/) (2015)
- [Kernel-based Reconstruction of Graph Signals](https://mlsanity.com/paper/1605.07174/) (2016)
- [Kernel-based Reconstruction of Space-time Functions on Dynamic Graphs](https://mlsanity.com/paper/1612.03615/) (2016)
- [Eigendecomposition-Free Sampling Set Selection for Graph Signals](https://mlsanity.com/paper/1809.01827/) (2018)
- [Graph Frequency Analysis of Brain Signals](https://mlsanity.com/paper/1512.00037/) (2015)
- [Kernel-Based Structural Equation Models for Topology Identification of Directed Networks](https://mlsanity.com/paper/1605.03122/) (2016)
- [Kernel-based Inference of Functions over Graphs](https://mlsanity.com/paper/1711.10353/) (2017)
- [Manifold Graph Signal Restoration using Gradient Graph Laplacian Regularizer](https://mlsanity.com/paper/2206.04245/) (2022)
- [Inference of Spatio-Temporal Functions over Graphs via Multi-Kernel Kriged Kalman Filtering](https://mlsanity.com/paper/1711.09306/) (2017)
- [Kernel Based Reconstruction for Generalized Graph Signal Processing](https://mlsanity.com/paper/2308.06949/) (2023)
- [Graph signal interpolation with Positive Definite Graph Basis Functions](https://mlsanity.com/paper/1912.02069/) (2019)
- [Multiple Kernel Representation Learning on Networks](https://mlsanity.com/paper/2106.05057/) (2021)
- [Fractional spectral graph wavelets and their applications](https://mlsanity.com/paper/1902.10471/) (2019)
- [Graph Signal Processing: Dualizing GSP Sampling in the Vertex and Spectral Domains](https://mlsanity.com/paper/2103.11020/) (2021)
- [Discrete Linear Canonical Transform on Graphs](https://mlsanity.com/paper/2209.12980/) (2022)

## Abstract (arXiv)

Graph signals are widely used to describe vertex attributes or features in graph-structured data, with applications spanning the internet, social media, transportation, sensor networks, and biomedicine. Graph signal processing (GSP) has emerged to facilitate the analysis, processing, and sampling of such signals. While kernel methods have been extensively studied for estimating graph signals from samples provided on a subset of vertices, their application to complex-valued graph signals remains largely unexplored. This paper introduces a novel framework for reconstructing graph signals using kernel methods on complex manifolds. By embedding graph vertices into a higher-dimensional complex ambient space that approximates a lower-dimensional manifold, the framework extends the reproducing kernel Hilbert space to complex manifolds. It leverages Hermitian metrics and geometric measures to characterize kernels and graph signals. Additionally, several traditional kernels and graph topology-driven kernels are proposed for reconstructing complex graph signals. Finally, experimental results on synthetic and real-world datasets demonstrate the effectiveness of this framework in accurately reconstructing complex graph signals, outperforming conventional kernel-based approaches. This work lays a foundational basis for integrating complex geometry and kernel methods in GSP.