Graph Learning From Data Under Laplacian and Structural Constraints

Abstract

Graphs are fundamental mathematical structures used in various fields to represent data, signals and processes. In this paper, we propose a novel framework for learning/estimating graphs from data. The proposed framework includes (i) formulation of various graph learning problems, (ii) their probabilistic interpretations and (iii) associated algorithms. Specifically, graph learning problems are posed as estimation of graph Laplacian matrices from some observed data under given structural constraints (e.g., graph connectivity and sparsity level). From a probabilistic perspective, the problems of interest correspond to maximum a posteriori (MAP) parameter estimation of Gaussian-Markov random field (GMRF) models, whose precision (inverse covariance) is a graph Laplacian matrix. For the proposed graph learning problems, specialized algorithms are developed by incorporating the graph Laplacian and structural constraints. The experimental results demonstrate that the proposed algorithms outperform the current state-of-the-art methods in terms of accuracy and computational efficiency.

Supervised learning and graph signal processing strategies for beam tracking in highly directional mobile communications

“…The goal of GSP is to analyze signals, defined over an irregular discrete domain, that are modeled over graphs. In the work of Egilmez et al, 17 a framework for learning and estimating graphs from raw data was proposed. 15,16 Time variance of real-world signals, which include those inherent to mobile communication systems, brings an interesting motivation to exploit signal processing techniques that deal with system dynamics.…”

“…Moreover, the parameter for the l 1 -regularization are tuned until some desired level of sparsity is reached, and according to the work of Egilmez et al, 17 the connectivity adjacency matrix A is set to represent a fully connected graph, ie, A = 11 T − I. Moreover, the parameter for the l 1 -regularization are tuned until some desired level of sparsity is reached, and according to the work of Egilmez et al, 17 the connectivity adjacency matrix A is set to represent a fully connected graph, ie, A = 11 T − I.…”

“…Similarly, the smaller edge weight, the larger probability that the difference between the same points is greater. The quadratic term x T Lx in (30) is commonly used to quantify the smoothness of graph signals 17 and is also known as the graph Laplacian quadratic form. 28 In this regard, the smaller Laplacian quadratic term indicates a smoother signal x.…”

Local stationarity of graph signals: insights and experiments

“…[10][11][12] We choose here the approach of Egilmez et al 10 for its ability to constraint the connectivity of the graph to any set of edges and learn their weights. Doing so, we show that the learnt graphs lead to smoother graph Fourier modes in the sense of the Euclidean domain, sparse graphs, and, more importantly, better stochastic models for the data in the sense of smoother graph power spectrum and more interpretable spectral components.…”

“…To learn the weights, we use the approach of Egilmez et al 10 In this work, the Laplacian matrix is optimized to make it coincide with the precision matrix Ω s = Σ † s = R † s of the data, with .…”