We study an extention of total variation denoising over images to over Cartesian power graphs and its applications to estimating non-parametric network models. The power graph fused lasso (PGFL) segments a matrix by exploiting a known graphical structure, @math , over the rows and columns. Our main results shows that for any connected graph, under subGaussian noise, the PGFL achieves the same mean-square error rate as 2D total variation denoising for signals of bounded variation. We study the use of the PGFL for denoising an observed network @math , where we learn the graph @math as the @math -nearest neighborhood graph of an estimated metric over the vertices. We provide theoretical and empirical results for estimating graphons, a non-parametric exchangeable network model, and compare to the state of the art graphon estimation methods.
Graph signal processing refers to methods that denoise, localize, detect, and predict signals over graphs. For example, each vertex corresponds to a low-powered sensor, and we would like to denoise sensor measurements, and we use the graph structure is based on communication between the sensors or spatial proximity. The driving assumption is that there is some underlying signal that in some way respects' the graph topology, and specifies the distribution of the observations. Many of the tools in signal processing and supervised learning can be extended to the graph case, such as Fourier analysis @cite_17 @cite_3 , wavelets @cite_10 @cite_34 @cite_8 , graph kernels @cite_32 , and convolutional networks @cite_13 @cite_19 . Graph structure has previously been used in matrix completion and network denoising problems (see for example, @cite_30 @cite_4 @cite_12 @cite_5 ), but these methods require some predetermined graph structure, such as knowledge graphs, so are not well suited to estimating graphons, and they do not perform segmentation, which is the focus of this work.
@cite_13 @cite_30 @cite_4 @cite_8 @cite_32 @cite_3 @cite_19 @cite_5 @cite_34 @cite_10 @cite_12 @cite_17
2030643321 2963702033 2030368230 2215668387
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. Many tools from the field of graph signal processing exploit knowledge of the underlying graph's structure (e.g., as encoded in the Laplacian matrix) to process signals on the graph. Therefore, in the case when no graph is available, graph signal processing tools cannot be used anymore. Researchers have proposed approaches to infer a graph topology from observations of signals on its vertices. Since the problem is ill-posed, these approaches make assumptions, such as smoothness of the signals on the graph, or sparsity priors. In this paper, we propose a characterization of the space of valid graphs, in the sense that they can explain stationary signals. To simplify the exposition in this paper, we focus here on the case where signals were i.i.d. at some point back in time and were observed after diffusion on a graph. We show that the set of graphs verifying this assumption has a strong connection with the eigenvectors of the covariance matrix, and forms a convex set. Along with a theoretical study in which these eigenvectors are assumed to be known, we consider the practical case when the observations are noisy, and experimentally observe how fast the set of valid graphs converges to the set obtained when the exact eigenvectors are known, as the number of observations grows. To illustrate how this characterization can be used for graph recovery, we present two methods for selecting a particular point in this set under chosen criteria, namely graph simplicity and sparsity. Additionally, we introduce a measure to evaluate how much a graph is adapted to signals under a stationarity assumption. Finally, we evaluate how state-of-the-art methods relate to this framework through experiments on a dataset of temperatures. 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. In many applications, the observations can be represented as a signal defined over the vertices of a graph. The analysis of such signals requires the extension of standard signal processing tools. In this paper, first, we provide a class of graph signals that are maximally concentrated on the graph domain and on its dual. Then, building on this framework, we derive an uncertainty principle for graph signals and illustrate the conditions for the recovery of band-limited signals from a subset of samples. We show an interesting link between uncertainty principle and sampling and propose alternative signal recovery algorithms, including a generalization to frame-based reconstruction methods. After showing that the performance of signal recovery algorithms is significantly affected by the location of samples, we suggest and compare a few alternative sampling strategies. Finally, we provide the conditions for perfect recovery of a useful signal corrupted by sparse noise, showing that this problem is also intrinsically related to vertex-frequency localization properties.