Signal Recovery on Graphs: Variation Minimization

Abstract

Abstract-We consider the problem of signal recovery on graphs. Graphs model data with complex structure as signals on a graph. Graph signal recovery recovers one or multiple smooth graph signals from noisy, corrupted, or incomplete measurements. We formulate graph signal recovery as an optimization problem, for which we provide a general solution through the alternating direction methods of multipliers. We show how signal inpainting, matrix completion, robust principal component analysis, and anomaly detection all relate to graph signal recovery and provide corresponding specific solutions and theoretical analysis. We validate the proposed methods on real-world recovery problems, including online blog classification, bridge condition identification, temperature estimation, recommender system for jokes, and expert opinion combination of online blog classification.

Crowdsource-based signal strength field estimation by Gaussian processes

“…We showed in the simulations that our approach is quite stable even when we introduce large errors in the locations of the users. Our method outperformed the ordinary Kriging, the inverse distance weighting methods, and GTVR from [4].…”

“…We considered 16 fixed uniformly placed nodes where we estimate the RSS and conducted 1000 different experiments. In all the simulations, we show the performance of our GP approach, the inverse distance weighting (IDW), the ordinary Kriging with detrending (OKD) technique and an the implementation of the graph signal inpainting via total variation regularization (GTVR) [4]. This last approach assumes smooth graph signals corrupted by noise and recovers the inaccessible graph signals from the accessible ones by minimizing the graph total variation based on the second norm.…”

Fractional graph-based semi-supervised learning

“…Finally, beyond the classification framework, it is very likely that Fractional G-SSL, in particular their differential-like version (γ > 1), can complement the recent attempts to apply G-SSL techniques to inpainting problems or more generally to signal recovery [16]. A first step towards this direction would be to cast Fractional G-SSL within the framework of Algebraic Graph Signal Processing as proposed in [17] or [18].…”

Detecting Localized Categorical Attributes on Graphs

“…It extends classical signal processing concepts such as signals, filters, Fourier transform, frequency response, low-and highpass filtering, from signals residing on regular lattices to data residing on general graphs; for example, a graph signal models the data value assigned to each node in a graph. Recent work involves sampling for graph signals [9], [10], [11], [12], recovery for graph signals [13], [14], [15], [16], representations for graph signals [17], [18] principles on graphs [19], [20], stationary graph signal processing [21], [22], graph dictionary construction [23], graph-based filter banks [24], [25], [26], [27], denoising on graphs [24], [28], community detection and clustering on graphs [29], [30], [31], distributed computing [32], [33] and graph-based transforms [34], [35], [36]. We here consider detecting localized categorical attributes on graphs.…”

“…The last inequality follows from (14). To simultaneously bound both type-1 and type-2 errors, the right hand side of (15) should be larger than the right hand side of (13).…”