Identifying graph topologies as well as processes evolving over graphs emerge in various applications involving gene-regulatory, brain, power, and social networks, to name a few. Key graph-aware learning tasks include regression, classification, subspace clustering, anomaly identification, interpolation, extrapolation, and dimensionality reduction. Scalable approaches to deal with such high-dimensional tasks experience a paradigm shift to address the unique modeling and computational challenges associated with data-driven sciences. Albeit simple and tractable, linear time-invariant models are limited since they are incapable of handling generally evolving topologies, as well as nonlinear and dynamic dependencies between nodal processes. To this end, the main goal of this paper is to outline overarching advances, and develop a principled framework to capture nonlinearities through kernels, which are judiciously chosen from a preselected dictionary to optimally fit the data. The framework encompasses and leverages (non) linear counterparts of partial correlation and partial Granger causality, as well as (non)linear structural equations and vector autoregressions, along with attributes such as low rank, sparsity, and smoothness to capture even directional dependencies with abrupt change points, as well as time-evolving processes over possibly time-evolving topologies. The overarching approach inherits the versatility and generality of kernel-based methods, and lends itself to batch and computationally affordable online learning algorithms, which include novel Kalman filters over graphs. Real data experiments highlight the impact of the nonlinear and dynamic models on consumer and financial networks, as well as gene-regulatory and functional connectivity brain networks, where connectivity patterns revealed exhibit discernible differences relative to existing approaches.
Bibliographical noteFunding Information:
Manuscript received September 25, 2017; revised January 5, 2018; accepted February 2, 2018. Date of current version April 24, 2018. This work was supported by the National Science Foundation (NSF) under Grants 1514056, 1500713, 1711471, and NIH 1R01GM104975-01. (Corresponding author: Georgios B. Giannakis.) The authors are with the Department of Electrical and Computer Engineering and the Digital Technology Center (DTC), University of Minnesota, Minneapolis, MN 55455 USA (e-mail: firstname.lastname@example.org; email@example.com; firstname.lastname@example.org).
This work was supported by the National Science Foundation (NSF) under Grants 1514056, 1500713, 1711471, and NIH 1R01GM104975-01.
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- Kernel-based models
- network topology inference
- nonlinear modeling
- time-varying networks