Abstract
A critical task in graph signal processing is to estimate the true signal from noisy observations over a subset of nodes, also known as the reconstruction problem. In this paper, we propose a node-adaptive regularization for graph signal reconstruction, which surmounts the conventional Tikhonov regularization, giving rise to more degrees of freedom; hence, an improved performance. We formulate the node-adaptive graph signal denoising problem, study its bias-variance trade-off, and identify conditions under which a lower mean squared error and variance can be obtained with respect to Tikhonov regularization. Compared with existing approaches, the node-adaptive regularization enjoys more general priors on the local signal variation, which can be obtained by optimally designing the regularization weights based on Prony's method or semidefinite programming. As these approaches require additional prior knowledge, we also propose a minimax (worst-case) strategy to address instances where this extra information is unavailable. Numerical experiments with synthetic and real data corroborate the proposed regularization strategy for graph signal denoising and interpolation, and show its improved performance compared with competing alternatives.
Highlights
Graphs, as models to represent data, do capture information about entities that comprise them, and encode the interactions between these entities
We propose a node-adaptive regularization for graph signal reconstruction, which surmounts the conventional Tikhonov regularization, giving rise to more degrees of freedom; an improved performance
CONTRIBUTIONS To be more specific, the main questions we address in this work are: (i) how the bias-variance trade-off behaves for the NA Tikhonov regularization? and (ii) how to design the NA weights optimally? Aimed to give answers to these questions, we make the following main contributions: 1) We formulate the NA Tikhonov regularization problem under a deterministic signal model assumption, derive its solution in closed-form, and study the respective bias-variance trade-off
Summary
As models to represent data, do capture information about entities (nodes) that comprise them, and encode the interactions between these entities (edges). In an attempt to increase the degrees of freedom (DoFs) of the regularization, in [15], it is proposed to minimize the signal total smoothness by regularizing separately the fitting error of each individual nodal measurement Despite that this approach can be considered as a multi-parameter based regularization, it only focuses on a measurement-wise regularization and ignores the coupling between the graph signal and the topology. Due to the need of a multi-parameter regularization, which considers the connectivity of the graph, in this paper, we propose a node-adaptive (NA) regularizer to increase the DoFs by applying node-dependent weights to fine-tune the trade-off between the fitting error and regularization term With these enhanced DoFs, we expect to achieve a better reconstruction performance without affecting the method complexity. 4) We corroborate the theoretical findings of this work, using both synthetic and real-world data, and show that the proposed NA denoising performs well with respect to the competing alternatives, especially in low signalto-noise ratio (SNR) settings
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