Graph Fourier Transform Based on ℓ_1 Norm Variation Minimization
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The definition of the graph Fourier transform is a fundamental issue in graph signal processing. Conventional graph Fourier transform is defined through the eigenvectors of the graph Laplacian matrix, which minimize the ℓ_2 norm signal variation. However, the computation of Laplacian eigenvectors is expensive when the graph is large. In this paper, we propose an alternative definition of graph Fourier transform based on the ℓ_1 norm variation minimization. We obtain a necessary condition satisfied by the ℓ_1 Fourier basis, and provide a fast greedy algorithm to approximate the ℓ_1 Fourier basis. Numerical experiments show the effectiveness of the greedy algorithm. Moreover, the Fourier transform under the greedy basis demonstrates a similar rate of decay to that of Laplacian basis for simulated or real signals.
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