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 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 1 norm variation minimization. We obtain a necessary condition satisfied by the 1 Fourier basis, and provide a fast greedy algorithm to approximate the 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.

This paper studies the convergence of the stochastic algorithm of the modified Robbins–Monro form for a Markov random field (MRF), in which some of the nodes are clamped to be observed variables while the others are hidden ones. Based on the theory of stochastic approximation, we propose proper assumptions to guarantee the Hölder regularity of both the update function and the solution of the Poisson equation. Under these assumptions, it is proved that the control parameter sequence is almost surely bounded and accordingly the algorithm converges to the stable point of the log-likelihood function with probability [Formula: see text].