Spectral Projector-Based Graph Fourier Transforms

This paper considers the definition of the graph Fourier transform (GFT) and of the spectral decomposition of graph signals. Current literature does not address the lack of unicity of the GFT. The GFT is the mapping from the signal set into its representation by a direct sum of irreducible shift invariant subspaces: 1) this decomposition may not be unique; and 2) there is freedom in the choice of basis for each component subspace. These issues are particularly relevant when the graph shift has repeated eigenvalues as is the case in many real-world applications; by ignoring them, there is no way of knowing if different researchers are using the same definition of the GFT and whether their results are comparable or not. The paper presents how to resolve the above degrees of freedom. We develop a quasi-coordinate free definition of the GFT and graph spectral decomposition of graph signals that we implement through oblique spectral projectors. We present properties of the GFT and of the spectral projectors and discuss a generalized Parseval's inequality. An illustrative example for a large real-world urban traffic dataset is provided.