Graph learning under sparsity priors

Many tools from the field of graph signal processing exploit knowledge of the underlying graph's structure (e.g., as encoded in the Laplacian matrix) to process signals on the graph. Therefore, in the case when no graph is available, graph signal processing tools cannot be used anymore. Researchers have proposed approaches to infer a graph topology from observations of signals on its nodes. Since the problem is ill-posed, these approaches make assumptions, such as smoothness of the signals on the graph, or sparsity priors. In this paper, we propose a characterization of the space of valid graphs, in the sense that they can explain stationary signals. To simplify the exposition in this paper, we focus here on the case where signals were i.i.d. at some point back in time and were observed after diffusion on a graph. We show that the set of graphs verifying this assumption has a strong connection with the eigenvectors of the covariance matrix, and forms a convex set. Along with a theoretical study in which these eigenvectors are assumed to be known, we consider the practical case when the observations are noisy, and experimentally observe how fast the set of valid graphs converges to the set obtained when the exact eigenvectors are known, as the number of observations grows. To illustrate how this characterization can be used for graph recovery, we present two methods for selecting a particular point in this set under chosen criteria, namely graph simplicity and sparsity. Additionally, we introduce a measure to evaluate how much a graph is adapted to signals under a stationarity assumption. Finally, we evaluate how state-of-the-art methods relate to this framework through experiments on a dataset of temperatures.

show abstract

“…Other popular matrices could be used instead to diffuse signals. For example, any polynome of T Ł could be used [13]- [15].…”

Section: Problem Formulation a Definitionsmentioning

confidence: 99%

“…In this section we review related work on reconstructing graphs from the observation of diffused signals and make connections to the approach we consider. Additional approaches exist but consider different signal models such as time series [13], [21], band-limited signals [22] or combinations of localized functions [14], [15].…”

Section: Related Workmentioning

confidence: 99%

Characterization and Inference of Graph Diffusion Processes From Observations of Stationary Signals

Pasdeloup

Gripon

Mercier

et al. 2018

IEEE Trans. on Signal and Inf. Process. over Networks

129

156

View full text Add to dashboard Cite

show abstract

“…2) Graph dictionary based learning frameworks: Methods belonging to this category are based on the notion of spectral graph dictionaries for efficient signal representation. Specifically, the authors in [47], [50] assume a different graph signal diffusion model, where the data consist of (sparse) combinations of overlapping local patterns that reside on the graph. These patterns may describe localized events or specific processes appearing at different vertices of the graph, such as traffic bottlenecks in transportation networks or rumor sources in social networks.…”

Section: ) Stationarity Based Learning Frameworkmentioning

confidence: 99%

Learning Graphs From Data: A Signal Representation Perspective

Dong

Thanou

Rabbat

et al. 2019

IEEE Signal Process. Mag.

Self Cite

322

217

View full text Add to dashboard Cite

The construction of a meaningful graph topology plays a crucial role in the effective representation, processing, analysis and visualization of structured data. When a natural choice of the graph is not readily available from the data sets, it is thus desirable to infer or learn a graph topology from the data. In this tutorial overview, we survey solutions to the problem of graph learning, including classical viewpoints from statistics and physics, and more recent approaches that adopt a graph signal processing (GSP) perspective. We further emphasize the conceptual similarities and differences between classical and GSP-based graph inference methods, and highlight the potential advantage of the latter in a number of theoretical and practical scenarios. We conclude with several open issues and challenges that are keys to the design of future signal processing and machine learning algorithms for learning graphs from data.

show abstract

“…Establishing connectivity among neurons, currently follows two main approaches, (i) Noise correlation analysis [8,9], often used in neuroscience to uncover the connectivity among neurons, (ii) Static graph learning [10,11] used to extract a fixed graph over time. When noise correlation is significant, it is commonly used in neuroscience as a short-time connectivity metric between every pair of neurons.…”

Section: Introductionmentioning

confidence: 99%

Dynamic Graph Learning: A Structure-Driven Approach

et al. 2021

View full text Add to dashboard Cite

The purpose of this paper is to infer a dynamic graph as a global (collective) model of time-varying measurements at a set of network nodes. This model captures both pairwise as well as higher order interactions (i.e., more than two nodes) among the nodes. The motivation of this work lies in the search for a connectome model which properly captures brain functionality across all regions of the brain, and possibly at individual neurons. We formulate it as an optimization problem, a quadratic objective functional and tensor information of observed node signals over short time intervals. The proper regularization constraints reflect the graph smoothness and other dynamics involving the underlying graph’s Laplacian, as well as the time evolution smoothness of the underlying graph. The resulting joint optimization is solved by a continuous relaxation of the weight parameters and an introduced novel gradient-projection scheme. While the work may be applicable to any time-evolving data set (e.g., fMRI), we apply our algorithm to a real-world dataset comprising recorded activities of individual brain cells. The resulting model is shown to be not only viable but also efficiently computable.

show abstract

Graph learning under sparsity priors

Cited by 24 publications

References 14 publications

Characterization and Inference of Graph Diffusion Processes From Observations of Stationary Signals

Characterization and Inference of Graph Diffusion Processes From Observations of Stationary Signals

Learning Graphs From Data: A Signal Representation Perspective

Dynamic Graph Learning: A Structure-Driven Approach

Contact Info

Product

Resources

About