Bandlimited Graph Signal Research Articles

Robust estimation of smooth graph signals from randomized space–time samples

Abstract Heat diffusion processes have found wide applications in modelling dynamical systems over graphs. In this paper, we consider the recovery of a $k$-bandlimited graph signal that is an initial signal of a heat diffusion process from its space–time samples. We propose three random space–time sampling regimes, termed dynamical sampling techniques, that consist in selecting a small subset of space–time nodes at random according to some probability distribution. We show that the number of space–time samples required to ensure stable recovery for each regime depends on a parameter called the spectral graph weighted coherence, which depends on the interplay between the dynamics over the graphs and sampling probability distributions. In optimal scenarios, as little as $\mathcal{O}(k \log (k))$ space–time samples are sufficient to ensure accurate and stable recovery of all $k$-bandlimited signals. Dynamical sampling typically requires much fewer spatial samples than the static case by leveraging the temporal information. Then, we propose a computationally efficient method to reconstruct $k$-bandlimited signals from their space–time samples. We prove that it yields accurate reconstructions and that it is also stable to noise. Finally, we test dynamical sampling techniques on a wide variety of graphs. The numerical results on synthetic and real climate datasets support our theoretical findings and demonstrate the efficiency.

Robust logarithmic hyperbolic cosine adaptive filtering over graph signals

Graph signal processing (GSP) has two methods including graph Fourier transform (GFT) and graph shift-operator (GSO) for dealing with the graph signal used for adaptive filtering (AF). However, in order to perform feature decomposition, GFT requires the availability of a band-limited graph signal with a known bandwidth in advance. Therefore, a general model based on GSO is then derived for AF by introducing a mixture coefficient to combine the advantages of different GSOs, comprehensively. Since the existing algorithms using the mean square error and other robust error criteria in GSP cannot effectively fight against non-Gaussian noises, a robust logarithmic hyperbolic cosine adaptive filter over graph (GLHCAF) algorithm is proposed by applying the logarithmic hyperbolic cosine loss function to the designed general model. The significant performance improvement of GLHCAF in non-Gaussian noises is obtained by simply controlling the scaling parameter. To avoid the parameter selections including scaling parameter and step size, the adaptive GLHCAF (AGLHCAF) algorithm is therefore proposed to further improve the performance of GLHCAF. Moreover, the mean square performance analysis of GLHCAF is given. Simulations in the real-world data are provided to assess the reliability and accuracy for theoretical analysis, the universality of the designed general model, and superiorities of proposed algorithms.

Near-Optimal Graph Signal Sampling by Pareto Optimization.

In this paper, we focus on the bandlimited graph signal sampling problem. To sample graph signals, we need to find small-sized subset of nodes with the minimal optimal reconstruction error. We formulate this problem as a subset selection problem, and propose an efficient Pareto Optimization for Graph Signal Sampling (POGSS) algorithm. Since the evaluation of the objective function is very time-consuming, a novel acceleration algorithm is proposed in this paper as well, which accelerates the evaluation of any solution. Theoretical analysis shows that POGSS finds the desired solution in quadratic time while guaranteeing nearly the best known approximation bound. Empirical studies on both Erdos-Renyi graphs and Gaussian graphs demonstrate that our method outperforms the state-of-the-art greedy algorithms.

Non-Bayesian Estimation Framework for Signal Recovery on Graphs

Graph signals arise from physical networks, such as power and communication systems, or as a result of a convenient representation of data with complex structure, such as social networks. We consider the problem of general graph signal recovery from noisy, corrupted, or incomplete measurements and under structural parametric constraints, such as smoothness in the graph frequency domain. In this paper, we formulate the graph signal recovery as a non-Bayesian estimation problem under a weighted mean-squared-error (WMSE) criterion, which is based on a quadratic form of the Laplacian matrix of the graph and its trace WMSE is the Dirichlet energy of the estimation error w.r.t. the graph. The Laplacian-based WMSE penalizes estimation errors according to their graph spectral content and is a difference-based cost function which accounts for the fact that in many cases signal recovery on graphs can only be achieved up to a constant addend. We develop a new Cram\'er-Rao bound (CRB) on the Laplacian-based WMSE and present the associated Lehmann unbiasedness condition w.r.t. the graph. We discuss the graph CRB and estimation methods for the fundamental problems of 1) A linear Gaussian model with relative measurements; and 2) Bandlimited graph signal recovery. We develop sampling allocation policies that optimize sensor locations in a network for these problems based on the proposed graph CRB. Numerical simulations on random graphs and on electrical network data are used to validate the performance of the graph CRB and sampling policies.

A-Optimal Sampling and Robust Reconstruction for Graph Signals via Truncated Neumann Series

Graph signal processing (GSP) studies signals that live on irregular data kernels described by graphs. One fundamental problem in GSP is sampling---from which subset of graph nodes to collect samples in order to reconstruct a bandlimited graph signal in high fidelity. In this paper, we seek a sampling strategy that minimizes the mean square error (MSE) of the reconstructed bandlimited graph signals assuming an independent and identically distributed (iid) noise model---leading naturally to the A-optimal design criterion. To avoid matrix inversion, we first prove that the inverse of the information matrix in the A-optimal criterion is equivalent to a Neumann matrix series. We then transform the truncated Neumann series based sampling problem into an equivalent expression that replaces eigenvectors of the Laplacian operator with a sub-matrix of an ideal low-pass graph filter. Finally, we approximate the ideal filter using a Chebyshev matrix polynomial. We design a greedy algorithm to iteratively minimize the simplified objective. For signal reconstruction, we propose an accompanied signal reconstruction strategy that reuses the approximated filter sub-matrix and is provably more robust than conventional least square recovery. Simulation results show that our sampling strategy outperforms two previous strategies in MSE performance at comparable complexity.

Bandlimited graph signal reconstruction by diffusion operator

Signal processing on graphs extends signal processing concepts and methodologies from the classical signal processing theory to data indexed by general graphs. For a bandlimited graph signal, the unknown data associated with unsampled vertices can be reconstructed from the sampled data by exploiting the spatial relationship of graph signal. In this paper, we propose a generalized analytical framework of unsampled graph signal and introduce a concept of diffusion operator which consists of local-mean and global-bias diffusion operator. Then, a diffusion operator-based iterative algorithm is proposed to reconstruct bandlimited graph signal from sampled data. In each iteration, the reconstructed residuals associated with the sampled vertices are diffused to all the unsampled vertices for accelerating the convergence. We then prove that the proposed reconstruction strategy converges to the original graph signal. The simulation results demonstrate the effectiveness of the proposed reconstruction strategy with various downsampling patterns, fluctuation of graph cut-off frequency, robustness on the classic graph structures, and noisy scenarios.

A Distributed Tracking Algorithm for Reconstruction of Graph Signals

The rapid development of signal processing on graphs provides a new perspective for processing large-scale data associated with irregular domains. In many practical applications, it is necessary to handle massive data sets through complex networks, in which most nodes have limited computing power. Designing efficient distributed algorithms is critical for this task. This paper focuses on the distributed reconstruction of a time-varying bandlimited graph signal based on observations sampled at a subset of selected nodes. A distributed least square reconstruction (DLSR) algorithm is proposed to recover the unknown signal iteratively, by allowing neighboring nodes to communicate with one another and make fast updates. DLSR uses a decay scheme to annihilate the out-of-band energy occurring in the reconstruction process, which is inevitably caused by the transmission delay in distributed systems. Proof of convergence and error bounds for DLSR are provided in this paper, suggesting that the algorithm is able to track time-varying graph signals and perfectly reconstruct time-invariant signals. The DLSR algorithm is numerically experimented with synthetic data and real-world sensor network data, which verifies its ability in tracking slowly time-varying graph signals.

Local-Set-Based Graph Signal Reconstruction

Signal processing on graph is attracting more and more attentions. For a graph signal in the low-frequency subspace, the missing data associated with unsampled vertices can be reconstructed through the sampled data by exploiting the smoothness of the graph signal. In this paper, the concept of local set is introduced and two local-set-based iterative methods are proposed to reconstruct bandlimited graph signal from sampled data. In each iteration, one of the proposed methods reweights the sampled residuals for different vertices, while the other propagates the sampled residuals in their respective local sets. These algorithms are built on frame theory and the concept of local sets, based on which several frames and contraction operators are proposed. We then prove that the reconstruction methods converge to the original signal under certain conditions and demonstrate the new methods lead to a significantly faster convergence compared with the baseline method. Furthermore, the correspondence between graph signal sampling and time-domain irregular sampling is analyzed comprehensively, which may be helpful to future works on graph signals. Computer simulations are conducted. The experimental results demonstrate the effectiveness of the reconstruction methods in various sampling geometries, imprecise priori knowledge of cutoff frequency, and noisy scenarios.