Graph Topology Learning and Signal Recovery Via Bayesian Inference

Ramezani-Mayiami, Mahmoud; Hajimirsadeghi, Mohammad; Skretting, Karl; Blum, Rick S.; Poor, H. Vincent

doi:10.1109/dsw.2019.8755601

Cited by 11 publications

(7 citation statements)

References 12 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The MMSE procedure is utilized to estimate the optimal Laplacian matrix, whose eigenvalue matrix is the precision matrix of the Gaussian Markov Random Field process. This work is an extension of our previous paper [54], in which we studied the problem of joint graph signal recovery and topology learning using an off-the-shelf optimization toolbox to implement the algorithm. However, in this version, we propose a fast algorithm and prove its convergence analytically.…”

Section: Introductionmentioning

confidence: 99%

Bayesian Topology Learning and noise removal from network data

Mayiami

Hajimirsadeghi

Skretting

et al. 2021

Discov Internet Things

Self Cite

View full text Add to dashboard Cite

Learning the topology of a graph from available data is of great interest in many emerging applications. Some examples are social networks, internet of things networks (intelligent IoT and industrial IoT), biological connection networks, sensor networks and traffic network patterns. In this paper, a graph topology inference approach is proposed to learn the underlying graph structure from a given set of noisy multi-variate observations, which are modeled as graph signals generated from a Gaussian Markov Random Field (GMRF) process. A factor analysis model is applied to represent the graph signals in a latent space where the basis is related to the underlying graph structure. An optimal graph filter is also developed to recover the graph signals from noisy observations. In the final step, an optimization problem is proposed to learn the underlying graph topology from the recovered signals. Moreover, a fast algorithm employing the proximal point method has been proposed to solve the problem efficiently. Experimental results employing both synthetic and real data show the effectiveness of the proposed method in recovering the signals and inferring the underlying graph.

show abstract

Section: Introductionmentioning

confidence: 99%

Bayesian Topology Learning and noise removal from network data

Mayiami

Hajimirsadeghi

Skretting

et al. 2021

Discov Internet Things

Self Cite

View full text Add to dashboard Cite

show abstract

“…GSP has a number of relevant applications, from spatiotemporal analysis of brain data [193]; to analyze vulnerabilities in power grid data [194]; to topological data analysis [195], chemoinformatics [196] and single cell transcriptomic analysis [197], to mention but a few examples. Statistical learning techniques have also being founded on a combination of MRFs and GSP [198,199], taking advantage of both the networked structure, the statistical dependence relationships and the temporal correlations of the signals [200][201][202]. Random field approaches to GSP have also been applied in the context of deep convolutional networks [203,204], often invoking features of the underlying joint conditional probability distributions such as ergodicity [205] and stationarity [206].…”

Section: Random Fields and Graph Signal Theorymentioning

confidence: 99%

Random Fields in Physics, Biology and Data Science

Hernández‐Lemus

2021

Front. Phys.

View full text Add to dashboard Cite

A random field is the representation of the joint probability distribution for a set of random variables. Markov fields, in particular, have a long standing tradition as the theoretical foundation of many applications in statistical physics and probability. For strictly positive probability densities, a Markov random field is also a Gibbs field, i.e., a random field supplemented with a measure that implies the existence of a regular conditional distribution. Markov random fields have been used in statistical physics, dating back as far as the Ehrenfests. However, their measure theoretical foundations were developed much later by Dobruschin, Lanford and Ruelle, as well as by Hammersley and Clifford. Aside from its enormous theoretical relevance, due to its generality and simplicity, Markov random fields have been used in a broad range of applications in equilibrium and non-equilibrium statistical physics, in non-linear dynamics and ergodic theory. Also in computational molecular biology, ecology, structural biology, computer vision, control theory, complex networks and data science, to name but a few. Often these applications have been inspired by the original statistical physics approaches. Here, we will briefly present a modern introduction to the theory of random fields, later we will explore and discuss some of the recent applications of random fields in physics, biology and data science. Our aim is to highlight the relevance of this powerful theoretical aspect of statistical physics and its relation to the broad success of its many interdisciplinary applications.

show abstract

“…The input graph signal, x, is shown to be smooth [14], i.e., its graph smoothness, as defined in (4), is small. Therefore, we model the distribution of the input graph signal, x, in the graph frequency domain [50], [51], as a smooth normal distribution, as follows:…”

Section: A Case Study: Psse In Electrical Networkmentioning

confidence: 99%

“…In (68) and (69) we use the notation that A S is the submatrix of A whose rows and columns are indicated by the set S, where S is the set of the remaining vertices. In addition, any sample-h-GSP estimator from (51), can be updated to the new topology for both cases: vertices added or removed, as follows:…”

Section: Example B: Estimation Under Topology Changesmentioning

confidence: 99%

Bayesian Estimation of Graph Signals

Kroizer,

Routtenberg,

Eldar

2021

Preprint

View full text Add to dashboard Cite

We consider the problem of recovering random graph signals from nonlinear measurements. For this case, closed-form Bayesian estimators are usually intractable and even numerical evaluation of these estimators may be hard to compute for large networks. In this paper, we propose a graph signal processing (GSP) framework for random graph signal recovery that utilizes the information of the structure behind the data. First, we develop the GSP-linear minimum mean-squared-error (GSP-LMMSE) estimator, which minimizes the mean-squarederror (MSE) among estimators that are represented as an output of a graph filter. The GSP-LMMSE estimator is based on diagonal covariance matrices in the graph frequency domain, and thus, has reduced complexity compared with the LMMSE estimator. This property is especially important when using the sample-mean versions of these estimators that are based on a training dataset. We then state conditions under which the low-complexity GSP-LMMSE estimator coincides with the optimal LMMSE estimator. Next, we develop the approximated parametrization of the GSP-LMMSE estimator by shift-invariant graph filters by solving a weighted least squared (WLS) problem. We present three implementations of the parametric GSP-LMMSE estimator for typical graph filters. Parametric graph filters are more robust to outliers and to network topology changes. In our simulations, we evaluate the performance of the proposed GSP-LMMSE estimators for the problem of state estimation in power systems, which can be interpreted as a graph signal recovery task. We show that the proposed sample-GSP estimators outperform the sample-LMMSE estimator for a limited training dataset and that the parametric GSP-LMMSE estimators are more robust to topology changes in the form of adding/removing vertices/edges.

show abstract

Graph Topology Learning and Signal Recovery Via Bayesian Inference

Cited by 11 publications

References 12 publications

Bayesian Topology Learning and noise removal from network data

Bayesian Topology Learning and noise removal from network data

Random Fields in Physics, Biology and Data Science

Bayesian Estimation of Graph Signals

Contact Info

Product

Resources

About