Predicting the future of Graph-supported Time Series (GTS) is a key challenge in many domains, such as climate monitoring, finance or neuroimaging. Yet it is a highly difficult problem as it requires to account jointly for time and graph (spatial) dependencies. To simplify this process, it is common to use a two-step procedure in which spatial and time dependencies are dealt with separately. In this paper, we are interested in comparing various linear spatial representations, namely structure-based ones and data-driven ones, in terms of how they help predict the future of GTS. To that end, we perform experiments with various datasets including spontaneous brain activity and raw videos.
When it comes to graphs, a natural approach to define a latent representation is a structure-based approach. It consists in defining the notion of frequency on a graph by analogy with the Fourier Transform (FT) @cite_2 . The so-called Graph Fourier Transform (GFT) represents the signal in another basis. This basis is built according to the graph structure without looking at the data at all. The eigenvectors of the Laplacian matrix of the graph are used as a new basis. These eigenvectors are ordered by increasing eigenvalues. Small eigenvalues correspond intuitively to low frequencies, and large eigenvalues to high frequencies. Consequently, to reduce the data dimension, only some of the first eigenvectors are kept. Data is back projected using only those first eigenvectors, and a low-dimensional representation of the signal is obtained. However, the analogy with the FT is not complete, as for instance, the GFT using a 2-dimensional grid does not match the 2D classical FT @cite_6 .
@cite_6 @cite_2
2963660991 2101491865
We propose an extension of Convolutional Neural Networks (CNNs) to graph-structured data, including strided convolutions and data augmentation defined from inferred graph translations. Our method matches the accuracy of state-of-the-art CNNs when applied on images, without any prior about their 2D regular structure. On fMRI data, we obtain a significant gain in accuracy compared with existing graph-based alternatives. In applications such as social, energy, transportation, sensor, and neuronal networks, high-dimensional data naturally reside on the vertices of weighted graphs. The emerging field of signal processing on graphs merges algebraic and spectral graph theoretic concepts with computational harmonic analysis to process such signals on graphs. In this tutorial overview, we outline the main challenges of the area, discuss different ways to define graph spectral domains, which are the analogs to the classical frequency domain, and highlight the importance of incorporating the irregular structures of graph data domains when processing signals on graphs. We then review methods to generalize fundamental operations such as filtering, translation, modulation, dilation, and downsampling to the graph setting and survey the localized, multiscale transforms that have been proposed to efficiently extract information from high-dimensional data on graphs. We conclude with a brief discussion of open issues and possible extensions.