Learning Laplacian Matrix in Smooth Graph Signal Representations

Abstract

The construction of a meaningful graph plays a crucial role in the success of many graph-based representations and algorithms for handling structured data, especially in the emerging field of graph signal processing. However, a meaningful graph is not always readily available from the data, nor easy to define depending on the application domain. In particular, it is often desirable in graph signal processing applications that a graph is chosen such that the data admit certain regularity or smoothness on the graph. In this paper, we address the problem of learning graph Laplacians, which is equivalent to learning graph topologies, such that the input data form graph signals with smooth variations on the resulting topology. To this end, we adopt a factor analysis model for the graph signals and impose a Gaussian probabilistic prior on the latent variables that control these signals. We show that the Gaussian prior leads to an efficient representation that favors the smoothness property of the graph signals. We then propose an algorithm for learning graphs that enforces such property and is based on minimizing the variations of the signals on the learned graph. Experiments on both synthetic and real world data demonstrate that the proposed graph learning framework can efficiently infer meaningful graph topologies from signal observations under the smoothness prior.

Graph adjacency matrix learning for irregularly sampled Markovian natural images

“…The reason why this occurs is because the graph tightly matches the actual signal f smoothness, since Finally, we test the proposed graph learning algorithm on a random graph, built by N = 25 nodes belonging to P communities, spatially distributed in equiangularly spaced circular regions with centers on the unit circle and unitary radius; we set the intra-community edge probability 0.8 and inter-community edge probability 0.2 and nonzero weights uniformly distributed in [0, 1]; the signal samples on nodes within communities are independently uniformly distributed in the community dependent ranges [p, p + 1] p = 1, · · ·P . We compare the graph learning results with those obtained by using the method GL-SigRep 2 in [1]. Specifically, Table III reports the F-measure, defined as the harmonic mean of the recall and precision performances of the proposed method and that in [1], averaged over 20 runs and for different values of P .…”

“…We compare the graph learning results with those obtained by using the method GL-SigRep 2 in [1]. Specifically, Table III reports the F-measure, defined as the harmonic mean of the recall and precision performances of the proposed method and that in [1], averaged over 20 runs and for different values of P . The ID-LD adjacency matrix well captures the signal smoothness, well performing w.r.t.…”

“…V (c) = V (|f i −f j |). Examples of such clique potential function can be found in the literature [5], and include, but are not limited to, quadratic 1 A clique is a set of pixels that belong to each other neighborhood.…”

“…of identifying the graph underlying the observed data values according to given criteria, such as graph smoothness or graph sparsity [1], [2]. In fact, except for specific SoG applications where the data are univocally associated to an underlying graph structure -e.g.…”

“…In mesh-based image representation, the graph is straightforwardly induced by the topology of the mesh [4], since the nodes and the signal samples at the nodes are extracted at the vertices of the mesh and the node links reproduce the mesh grid; this is exemplified in Figs.1,2(a)-(c), where two different images, a depth map and a nevus image, are displayed, together with the meshes obtained by Delaunay triangulation and the corresponding graphs. Model-based graph learning has been recently analyzed in [1], where the authors resort to a parametric smooth signal model and develop a procedure for learning the graph Laplacian matrix via an alternating minimization algorithm jointly enforcing the SoG smoothness and the Laplacian properties of sparsity and semidefinite positiveness. In [2], the authors propose an iterative Laplacian matrix learning algorithm that, stemming on the knowledge of which edges are active, updates one row/column of the precision matrix at a time by solving a non-negative quadratic program and superimposing prior constraints on the Laplacian matrix structure.…”