# Multi-dimensional Graph Fourier Transform

arXiv: https://arxiv.org/abs/1712.07811 · HTML: https://mlsanity.com/paper/1712.07811/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 21 Dec 2017.

## Built on by (most cited first, 5 of 5)

- [A Multidimensional Graph Fourier Transformation Neural Network for Vehicle Trajectory Prediction](https://mlsanity.com/paper/2305.07416/) (2023)
- [Prediction and Interpretation of Vehicle Trajectories in the Graph Spectral Domain](https://mlsanity.com/paper/2309.16702/) (2023)
- [Beyond Spatio-Temporal Representations: Evolving Fourier Transform for Temporal Graphs](https://mlsanity.com/paper/2402.16078/) (2024)
- [Beyond 5G Network Failure Classification for Network Digital Twin Using Graph Neural Network](https://mlsanity.com/paper/2406.06595/) (2024)
- [Graph Neural Network-based Spectral Filtering Mechanism for Imbalance Classification in Network Digital Twin](https://mlsanity.com/paper/2502.11505/) (2025)

## Abstract (arXiv)

Many signals on Cartesian product graphs appear in the real world, such as digital images, sensor observation time series, and movie ratings on Netflix. These signals are "multi-dimensional" and have directional characteristics along each factor graph. However, the existing graph Fourier transform does not distinguish these directions, and assigns 1-D spectra to signals on product graphs. Further, these spectra are often multi-valued at some frequencies. Our main result is a multi-dimensional graph Fourier transform that solves such problems associated with the conventional GFT. Using algebraic properties of Cartesian products, the proposed transform rearranges 1-D spectra obtained by the conventional GFT into the multi-dimensional frequency domain, of which each dimension represents a directional frequency along each factor graph. Thus, the multi-dimensional graph Fourier transform enables directional frequency analysis, in addition to frequency analysis with the conventional GFT. Moreover, this rearrangement resolves the multi-valuedness of spectra in some cases. The multi-dimensional graph Fourier transform is a foundation of novel filterings and stationarities that utilize dimensional information of graph signals, which are also discussed in this study. The proposed methods are applicable to a wide variety of data that can be regarded as signals on Cartesian product graphs. This study also notes that multivariate graph signals can be regarded as 2-D univariate graph signals. This correspondence provides natural definitions of the multivariate graph Fourier transform and the multivariate stationarity based on their 2-D univariate versions.