Learning sparse graphs under smoothness prior

Abstract

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.

Connecting the Dots: Identifying Network Structure via Graph Signal Processing

“…Importantly, (10) together with (5) . The neighborhood-based lasso method in [36] cycles over vertices i = 1, .…”

“…, x P ] ∈ R N ×P are assumed to vary smoothly on the sparse graph G and the actual number of edges |E| is assumed to be considerably smaller than the maximum possible number of edges M := N 2 = N (N − 1)/2. We describe the approach in [5], whose idea is to cast the graph learning problem as one of edge subset selection. As we show in the sequel, it is possible to parametrize the unknown graph topology via a sparse edge selection vector.…”

“…We can now formally state the problem studied in [5]. Given graph signals X := {x p } P p=1 , determine an undirected and unweighted graph G with K edges such that the signals in X exhibit smooth variations on G. A natural formulation given the model presented so far, is to solve the optimization problem…”

Graph Error Effect in Graph Signal Processing

“…In GSP tasks such as filtering or signal recovery from samples, smoothness with respect to the chosen shift matrix is a key assumption. Therefore, smoothness is used also in the shift operator learning from training data, as for example in [13]. Graph autocorrelation is a measure of smoothness which we will discuss here, since in the next section in the context of independent component analysis, we deal with a related concept for multivariate signals, graph autocorrelation matrix.…”

“…The derivations of the expected values in the other two equations in (10) are more tedious and a lot lengthier, but otherwise follow the same steps as the derivation of (13). Thus, we skip these derivations here for the sake of brevity, and write down the solution for the system of equations (10).…”