  In this paper, we are interested in learning the underlying graph structure
behind training data. Solving this basic problem is essential to carry out any
graph signal processing or machine learning task. To realize this, we assume
that the data is smooth with respect to the graph topology, and we parameterize
the graph topology using an edge sampling function. That is, the graph
Laplacian is expressed in terms of a sparse edge selection vector, which
provides an explicit handle to control the sparsity level of the graph. We
solve the sparse graph learning problem given some training data in both the
noiseless and noisy settings. Given the true smooth data, the posed sparse
graph learning problem can be solved optimally and is based on simple rank
ordering. Given the noisy data, we show that the joint sparse graph learning
and denoising problem can be simplified to designing only the sparse edge
selection vector, which can be solved using convex optimization.
