LG ] 12 S ep 2 01 6 LEARNING SPARSE GRAPHS UNDER SMOOTHNESS PRIOR

In this paper, we are interested in learning the underlying g raph structure behind training data. Solving this basic problem is essential to carry out any graph signal processing or machine lear ning task. To realize this, we assume that the data is smooth with r espect to the graph topology, and we parameterize the graph topolog y using an edge sampling function. That is, the graph Laplacian is ex pressed in terms of a sparse edge selection vector, which provides an xplicit handle to control the sparsity level of the graph. We solve th e sparse graph learning problem given some training data in both the n oiseless and noisy settings. Given the true smooth data, the pose d sparse graph learning problem can be solved optimally and is based o n simple rank ordering. Given the noisy data, we show that the join t sparse graph learning and denoising problem can be simplified to des igning only the sparse edge selection vector, which can be solve d using convex optimization.