State-Space Network Topology Identification from Partial Observations

In this work, we explore the state-space formulation of a network process to recover, from partial observations, the underlying network topology that drives its dynamics. To do so, we employ subspace techniques borrowed from system identification literature and extend them to the network topology identification problem. This approach provides a unified view of the traditional network control theory and signal processing on graphs. In addition, it provides theoretical guarantees for the recovery of the topological structure of a deterministic continuous-time linear dynamical system from input-output observations even though the input and state interaction networks might be different. The derived mathematical analysis is accompanied by an algorithm for identifying, from data, a network topology consistent with the dynamics of the system and conforms to the prior information about the underlying structure. The proposed algorithm relies on alternating projections and is provably convergent. Numerical results corroborate the theoretical findings and the applicability of the proposed algorithm.Comment: 13 pages, 3 appendix page

Graph Signal Processing: Overview, Challenges and Applications

Research in Graph Signal Processing (GSP) aims to develop tools for processing data defined on irregular graph domains. In this paper we first provide an overview of core ideas in GSP and their connection to conventional digital signal processing. We then summarize recent developments in developing basic GSP tools, including methods for sampling, filtering or graph learning. Next, we review progress in several application areas using GSP, including processing and analysis of sensor network data, biological data, and applications to image processing and machine learning. We finish by providing a brief historical perspective to highlight how concepts recently developed in GSP build on top of prior research in other areas.Comment: To appear, Proceedings of the IEE

Graph-based learning under perturbations via total least-squares

Graphs are pervasive in different fields unveiling complex relationships between data. Two major graph-based learning tasks are topology identification and inference of signals over graphs. Among the possible models to explain data interdependencies, structural equation models (SEMs) accommodate a gamut of applications involving topology identification. Obtaining conventional SEMs though requires measurements across nodes. On the other hand, typical signal inference approaches “blindly trust” a given nominal topology. In practice however, signal or topology perturbations may be present in both tasks, due to model mismatch, outliers, outages or adversarial behavior. To cope with such perturbations, this work introduces a regularized total least-squares (TLS) approach and iterative algorithms with convergence guarantees to solve both tasks. Further generalizations are also considered relying on structured and/or weighted TLS when extra prior information on the perturbation is available. Analyses with simulated and real data corroborate the effectiveness of the novel TLS-based approaches

Online Joint Topology Identification and Signal Estimation with Inexact Proximal Online Gradient Descent

Identifying the topology that underlies a set of time series is useful for tasks such as prediction, denoising, and data completion. Vector autoregressive (VAR) model based topologies capture dependencies among time series, and are often inferred from observed spatio-temporal data. When the data are affected by noise and/or missing samples, the tasks of topology identification and signal recovery (reconstruction) have to be performed jointly. Additional challenges arise when i) the underlying topology is time-varying, ii) data become available sequentially, and iii) no delay is tolerated. To overcome these challenges, this paper proposes two online algorithms to estimate the VAR model-based topologies. The proposed algorithms have constant complexity per iteration, which makes them interesting for big data scenarios. They also enjoy complementary merits in terms of complexity and performance. A performance guarantee is derived for one of the algorithms in the form of a dynamic regret bound. Numerical tests are also presented, showcasing the ability of the proposed algorithms to track the time-varying topologies with missing data in an online fashion.Comment: 14 pages including supplementary material, 2 figures, submitted to IEEE Transactions on Signal Processin

Kernel-based graph learning from smooth signals: a functional viewpoint

The problem of graph learning concerns the construction of an explicit topological structure revealing the relationship between nodes representing data entities, which plays an increasingly important role in the success of many graph-based representations and algorithms in the field of machine learning and graph signal processing. In this paper, we propose a novel graph learning framework that incorporates prior information along node and observation side, and in particular the covariates that help to explain the dependency structures in graph signals. To this end, we consider graph signals as functions in the reproducing kernel Hilbert space associated with a Kronecker product kernel, and integrate functional learning with smoothness-promoting graph learning to learn a graph representing the relationship between nodes. The functional learning increases the robustness of graph learning against missing and incomplete information in the graph signals. In addition, we develop a novel graph-based regularisation method which, when combined with the Kronecker product kernel, enables our model to capture both the dependency explained by the graph and the dependency due to graph signals observed under different but related circumstances, e.g. different points in time. The latter means the graph signals are free from the i.i.d. assumptions required by the classical graph learning models. Experiments on both synthetic and real-world data show that our methods outperform the state-of-the-art models in learning a meaningful graph topology from graph signals, in particular with heavy noise, missing values, and multiple dependency

Kernel-based Graph Learning from Smooth Signals: A Functional Viewpoint

The problem of graph learning concerns the construction of an explicit topological structure revealing the relationship between nodes representing data entities, which plays an increasingly important role in the success of many graph-based representations and algorithms in the field of machine learning and graph signal processing. In this paper, we propose a novel graph learning framework that incorporates the node-side and observation-side information, and in particular the covariates that help to explain the dependency structures in graph signals. To this end, we consider graph signals as functions in the reproducing kernel Hilbert space associated with a Kronecker product kernel, and integrate functional learning with smoothness-promoting graph learning to learn a graph representing the relationship between nodes. The functional learning increases the robustness of graph learning against missing and incomplete information in the graph signals. In addition, we develop a novel graph-based regularisation method which, when combined with the Kronecker product kernel, enables our model to capture both the dependency explained by the graph and the dependency due to graph signals observed under different but related circumstances, e.g. different points in time. The latter means the graph signals are free from the i.i.d. assumptions required by the classical graph learning models. Experiments on both synthetic and real-world data show that our methods outperform the state-of-the-art models in learning a meaningful graph topology from graph signals, in particular under heavy noise, missing values, and multiple dependency.Comment: 13 pages, with extra 3-page appendice