Enhanced lasso recovery on graph

Abstract

This work aims at recovering signals that are sparse on graphs. Compressed sensing offers techniques for signal recovery from a few linear measurements and graph Fourier analysis provides a signal representation on graph. In this paper, we leverage these two frameworks to introduce a new Lasso recovery algorithm on graphs. More precisely, we present a non-convex, non-smooth algorithm that outperforms the standard convex Lasso technique. We carry out numerical experiments on three benchmark graph datasets. Sparse Representation on GraphsThe goal of this work is to reconstruct signals on graphs that are supposed to be sparse in the graph Fourier representation. In this context, we will deal here with two main concepts, graph and sparsity, which have gathered a lot of attention in the recent years with the emergence of Compressed Sensing and Big Data. Let us introduce briefly these two concepts in the rest of this section.Graph/network is a powerful tool to represent complex high-dimensional datasets, in the sense that a graph structures data with respect to their similarities. Graphs have become increasingly more considered in applications such as search engines, social networks, airline routes, 3D geometric shapes, human brain connectivity, etc. Mathematics offer strong theoretical tools to analyze graphs with Harmonic Analysis and Spectral Graph Theory. An essential graph analysis tool is the graph Laplacian operator, which is the discrete approximation of the continuum Laplace-Beltrami operator for smooth manifolds. It is known that the eigenvectors of the Laplace-Beltrami operator provide a local parametrization of the manifold [1]. Equivalently, the eigenvectors of the graph Laplacian, also called graph Fourier modes, provides a representation of the graph. Given a graph with (V, E, W ), V , E and W being respectively the set of n nodes, the set of edges and the similarity/adjacency matrix, then the (unnormalized) graph Laplacian operator is defined aswhere D is the diagonal degree matrix s.t. is its Fourier transform. In this paper, we consider three well-known graphs. First, the synthetic LFR graph, which was introduced in [2] to study community graphs. Here, the number of nodes is chosen to be n = 1, 000, the number of communities is 10 and the degree of community overlapping is µ = 0.4. Second, we consider a coarse version (for computational speedup) of the benchmark MNIST dataset of NYU [3] with n = 1, 176 nodes and the number of classes is 10. Last, we use a coarse version of the well-known 20newsgroups dataset of CMU [4] with n = 1, 432 nodes and the number of classes is 20. All three dataset graphs are illustrated on Figure 1 with their graph Laplacian spectrum.

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