Distributed Adaptive Learning of Graph Signals

The aim of this paper is to propose distributed strategies for adaptive learning of signals defined over graphs. Assuming the graph signal to be bandlimited, the method enables distributed reconstruction, with guaranteed performance in terms of mean-square error, and tracking from a limited number of sampled observations taken from a subset of vertices. A detailed mean square analysis is carried out and illustrates the role played by the sampling strategy on the performance of the proposed method. Finally, some useful strategies for distributed selection of the sampling set are provided. Several numerical results validate our theoretical findings, and illustrate the performance of the proposed method for distributed adaptive learning of signals defined over graphs.Comment: To appear in IEEE Transactions on Signal Processing, 201

Sampling of time-varying network signals from equation-driven to data-driven techniques

Sampling and recovering the time-varying network signals via the subset of network vertices is essential for a wide range of scientific and engineering purposes. Current studies on sampling a single (continuous) time-series or a static network data, are not suitable for time-varying network signals. This will be even more challenging when there is a lack of explicit dynamic models and signal-space that indicate the time-evolution and vertex dependency. The work begins by bridging the time-domain sampling frequency and the network-domain sampling vertices, via the eigenvalues of the graph Fourier transform (GFT) operator composed by the combined dynamic equations and network topology. Then, for signals with hidden governing mechanisms, we propose a data-driven GFT sampling method using a prior signal-space. We characterize the signal dependency (among vertices) into the graph bandlimited frequency domain, and map such bandlimitedness into optimal sampling vertices. Furthermore, to achieve dynamic model and signal-space independent sensor placement, a Koopman based nonlinear GFT sampling is proposed. A novel data-driven Log-Koopman operator is designed to extract a linearized evolution model using small (M = O(N)) and decoupled observables defined on N original vertices. Then, nonlinear GFT is proposed to derive sampling vertices, by exploiting the inherent nonlinear dependence between M observables (defined on N < M vertices), and the time-evolved information presented by Log-Koopman evolution model. The work also informs the planned future work to formulate an easy-to-use and explainable neural network (NN) based sampling framework, for real-world industrial engineering and applications

Approximating Spectral Clustering via Sampling: a Review

Spectral clustering refers to a family of unsupervised learning algorithms that compute a spectral embedding of the original data based on the eigenvectors of a similarity graph. This non-linear transformation of the data is both the key of these algorithms' success and their Achilles heel: forming a graph and computing its dominant eigenvectors can indeed be computationally prohibitive when dealing with more that a few tens of thousands of points. In this paper, we review the principal research efforts aiming to reduce this computational cost. We focus on methods that come with a theoretical control on the clustering performance and incorporate some form of sampling in their operation. Such methods abound in the machine learning, numerical linear algebra, and graph signal processing literature and, amongst others, include Nystr\"om-approximation, landmarks, coarsening, coresets, and compressive spectral clustering. We present the approximation guarantees available for each and discuss practical merits and limitations. Surprisingly, despite the breadth of the literature explored, we conclude that there is still a gap between theory and practice: the most scalable methods are only intuitively motivated or loosely controlled, whereas those that come with end-to-end guarantees rely on strong assumptions or enable a limited gain of computation time

Sampling and inference of networked dynamics using Log-Koopman nonlinear graph fourier transform

Monitoring the networked dynamics via the subset of nodes is essential for a variety of scientific and operational purposes. When there is a lack of an explicit model and networked signal space, traditional observability analysis and non-convex methods are insufficient. Current data-driven Koopman linearization, although derives a linear evolution model for selected vector-valued observable of original state-space, may result in a large sampling set due to: (i) the large size of polynomial based observables (O(N2) , N number of nodes in network), and (ii) not factoring in the nonlinear dependency betweenobservables. In this work, to achieve linear scaling (O(N) ) and a small set of sampling nodes, wepropose to combine a novel Log-Koopman operator and nonlinear Graph Fourier Transform (NL-GFT) scheme. First, the Log-Koopman operator is able to reduce the size of observables by transforming multiplicative poly-observable to logarithm summation. Second, anonlinear GFT concept and sampling theory are provided to exploit the nonlinear dependence of observables for observability analysis using Koopman evolution model. The results demonstrate that the proposed Log-Koopman NL-GFT scheme can (i) linearize unknownnonlinear dynamics using O(N) observables, and (ii) achieve lower number of sampling nodes, compared with the state-of-the art polynomial Koopman based observability analysis

Applied Harmonic Analysis and Data Processing

Massive data sets have their own architecture. Each data source has an inherent structure, which we should attempt to detect in order to utilize it for applications, such as denoising, clustering, anomaly detection, knowledge extraction, or classification. Harmonic analysis revolves around creating new structures for decomposition, rearrangement and reconstruction of operators and functions—in other words inventing and exploring new architectures for information and inference. Two previous very successful workshops on applied harmonic analysis and sparse approximation have taken place in 2012 and in 2015. This workshop was the an evolution and continuation of these workshops and intended to bring together world leading experts in applied harmonic analysis, data analysis, optimization, statistics, and machine learning to report on recent developments, and to foster new developments and collaborations