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.@en

We study the sensor selection problem for field estimation, where a best subset of sensors is activated to monitor a spatially correlated random field. Different from most commonly used centralized selection algorithms, we propose a decentralized architecture where sensor selection can be carried out in a distributed way and by the sensors themselves. A decentralized approach is essential since each sensor has access only to the information (e.g., correlation) in its neighborhood. To make distributed optimization possible, we decompose the global cost function into local cost functions that require only the information in local neighborhoods of sensors. We then employ the alternating direction method of multipliers (ADMM) to solve the proposed sensor selection problem. In our algorithm, each sensor solves small-scale optimization problems, and communicates directly only with its immediate neighbors. Numerical results are provided to show the effectiveness of our approach.

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