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.
Some theoretical results have been established for the sampling problem of bandlimited graph-based signals; see e.g., @cite_30 @cite_24 @cite_20 . The relation between the sample size necessary to obtain unique reconstruction and the cutoff frequency of bandlimited signal space has been studied. Similar to classical results on time-domain irregular sampling, the idea of frame" has been introduced for graph signal processing. The unique reconstruction conditions have been derived for normalized and unnormalized Laplacians @cite_30 @cite_43 . As the field of graph signal processing is rapidly developing, we summarize some recent related works as follows. A least square approach has been proposed in @cite_27 to reconstruct bandlimited graph signal from signal values observed on sampled vertices, using a centralized algorithm. An iterative method of bandlimited graph signal reconstruction has been proposed in @cite_45 , with the practical consideration of balancing a tradeoff between smoothness and data-fitting. Two more efficient iterative reconstruction methods using the local set have been considered in @cite_2 @cite_38 . A necessary and sufficient condition for perfect reconstruction of bandlimited graph signal has been derived in @cite_0 . Readers may refer to Section for more details of related works.
@cite_30 @cite_38 @cite_24 @cite_43 @cite_27 @cite_45 @cite_0 @cite_2 @cite_20
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A notion of Paley-Wiener spaces on combinatorial graphs is introduced. It is shown that functions from some of these spaces are uniquely determined by their values on some sets of vertices which are called the uniqueness sets. Such uniqueness sets are described in terms of Poincare-Wirtinger-type inequalities. A reconstruction algorithm of Paley-Wiener functions from uniqueness sets which uses the idea of frames in Hilbert spaces is developed. Special consideration is given to the n-dimensional lattice, homogeneous trees, and eigenvalue and eigenfunction problems on finite graphs. 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. Notions of interpolating variational splines and Paley–Wiener spaces are introduced on a combinatorial graph G. Both of these definitions explore existence of a combinatorial Laplace operator on G. The existence and uniqueness of interpolating variational splines on a graph is shown. As an application of variational splines, the paper presents a reconstruction algorithm of Paley–Wiener functions on graphs from their uniqueness sets. In this paper, we propose a novel algorithm to interpolate data defined on graphs, using signal processing concepts. The interpolation of missing values from known samples appears in various applications, such as matrix vector completion, sampling of high-dimensional data, semi-supervised learning etc. In this paper, we formulate the data interpolation problem as a signal reconstruction problem on a graph, where a graph signal is defined as the information attached to each node (scalar or vector values mapped to the set of vertices edges of the graph). We use recent results for sampling in graphs to find classes of bandlimited (BL) graph signals that can be reconstructed from their partially observed samples. The interpolated signal is obtained by projecting the input signal into the appropriate BL graph signal space. Additionally, we impose a bilateral' weighting scheme on the links between known samples, which further improves accuracy. We use our proposed method for collaborative filtering in recommendation systems. Preliminary results show a very favorable trade-off between accuracy and complexity, compared to state of the art algorithms. In this paper, we present two localized graph filtering based methods for interpolating graph signals defined on the vertices of arbitrary graphs from only a partial set of samples. The first method is an extension of previous work on reconstructing bandlimited graph signals from partially observed samples. The iterative graph filtering approach very closely approximates the solution proposed in the that work, while being computationally more efficient. As an alternative, we propose a regularization based framework in which we define the cost of reconstruction to be a combination of smoothness of the graph signal and the reconstruction error with respect to the known samples, and find solutions that minimize this cost. We provide both a closed form solution and a computationally efficient iterative solution of the optimization problem. The experimental results on the recommendation system datasets demonstrate effectiveness of the proposed methods. In this paper, we extend the Nyquist-Shannon theory of sampling to signals defined on arbitrary graphs. Using spectral graph theory, we establish a cut-off frequency for all bandlimited graph signals that can be perfectly reconstructed from samples on a given subset of nodes. The result is analogous to the concept of Nyquist frequency in traditional signal processing. We consider practical ways of computing this cut-off and show that it is an improvement over previous results. We also propose a greedy algorithm to search for the smallest possible sampling set that guarantees unique recovery for a signal of given bandwidth. The efficacy of these results is verified through simple examples. Signal processing on graph is attracting more and more attention. For a graph signal in the low-frequency subspace, the missing data on the vertices of graph can be reconstructed through the sampled data by exploiting the smoothness of graph signal. In this paper, two iterative methods are proposed to reconstruct bandlimited graph signal from sampled data. In each iteration, one of the proposed methods weights the sampled residual for different vertices, while the other conducts a limited propagation operation. Both the methods are proved to converge to the original signal under certain conditions. The proposed methods lead to a significantly faster convergence compared with the baseline method. Experiment results of synthetic graph signal and the real world data demonstrate the effectiveness of the reconstruction methods.