Graph Topology Inference Based on Sparsifying Transform Learning

Abstract

Graph-based representations play a key role in machine learning. The fundamental step in these representations is the association of a graph structure to a dataset. In this paper, we propose a method that aims at finding a block sparse representation of the graph signal leading to a modular graph whose Laplacian matrix admits the found dictionary as its eigenvectors. The role of sparsity here is to induce a bandlimited representation or, equivalently, a modular structure of the graph. The proposed strategy is composed of two optimization steps: i) learning an orthonormal sparsifying transform from the data; ii) recovering the Laplacian, and then topology, from the transform. The first step is achieved through an iterative algorithm whose alternating intermediate solutions are expressed in closed form. The second step recovers the Laplacian matrix from the sparsifying transform through a convex optimization method. Numerical results corroborate the effectiveness of the proposed methods over both synthetic data and real brain data, used for inferring the brain functionality network through experiments conducted over patients affected by epilepsy.

Bayesian Topology Learning and noise removal from network data

“…Some other research works with similar ideas are presented for different stationary and non-stationary processes in [26][27][28][29][30]. Dictionary learning [31,32] and transform learning [33] have also been used for inferring the graph topology. In [31][32][33], a specific relation between the Laplacian matrix and the dictionary atoms has been sought and hence these algorithms are applicable when we have some knowledge about signal representation in Fig.…”

“…Dictionary learning [31,32] and transform learning [33] have also been used for inferring the graph topology. In [31][32][33], a specific relation between the Laplacian matrix and the dictionary atoms has been sought and hence these algorithms are applicable when we have some knowledge about signal representation in Fig. 1 Illustration of learning a network graph structure from IoT data 1 The GSO is a matrix which captures the graph's local topology and the graph Fourier transform is defined using its eigenvectors.…”

Spread of Information With Confirmation Bias in Cyber-Social Networks

“…For the directed communication graph with n agents, there are n 2 −n weighted communication links (variables) [L] ij , i = j, need to be inferred. In (7), both matricesLU + UL ⊤ and V are symmetric. Thus, (7) contains, at most, (n+1)n 2 distinct linear equations that are related to one or some [L] ij , i = j.…”

“…In (7), both matricesLU + UL ⊤ and V are symmetric. Thus, (7) contains, at most, (n+1)n 2 distinct linear equations that are related to one or some [L] ij , i = j. Moreover, we note that the constraint conditionsL1 n = 1 n and [L] ij ≥ 0 in (7) only reduce the number of linear equations while do not affect the number of variables to be inferred.…”

Graph Learning Under Spectral Sparsity Constraints

“…Based on this model and criterion, we proceed in two phases -we first provide an algorithm for estimating the eigenvectors of the graph matrix and then proceed to find the eigenvalues from the eigenvectors in the second phase [9]. In principle, our model in this paper is similar to the block sparsity models of [16], [17] where all signals are assumed to have the same frequency support and behave smoothly with respect to the underlying graph structure; however, the crucial difference is that we allow our data to have arbitrary frequency support anywhere within the entire frequency range. We refer to these as wideband graph signals.…”