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.
We propose a novel recovery algorithm for signals with complex, irregular structure that is commonly represented by graphs. Our approach is a generalization of the signal inpainting technique from classical signal processing. We formulate corresponding minimization problems and demonstrate that in many cases they have closed-form solutions. We discuss a relation of the proposed approach to regression, provide an upper bound on the error for our algorithm and compare the proposed technique with other existing algorithms on real-world datasets. We present an adaptive graph filtering approach to semi-supervised classification. Adaptive graph filters combine decisions from multiple graph filters using a weighting function that is optimized in a semi-supervised manner. We also demonstrate the multiresolution property of adaptive graph filters by connecting them to the diffusion wavelets. In our experiments, we apply the adaptive graph filters to the classification of online blogs and damage identification in indirect bridge structural health monitoring. 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 social settings, individuals interact through webs of relationships. Each individual is a node in a complex network (or graph) of interdependencies and generates data, lots of data. We label the data by its source, or formally stated, we index the data by the nodes of the graph. The resulting signals (data indexed by the nodes) are far removed from time or image signals indexed by well ordered time samples or pixels. DSP, discrete signal processing, provides a comprehensive, elegant, and efficient methodology to describe, represent, transform, analyze, process, or synthesize these well ordered time or image signals. This paper extends to signals on graphs DSP and its basic tenets, including filters, convolution, z-transform, impulse response, spectral representation, Fourier transform, frequency response, and illustrates DSP on graphs by classifying blogs, linear predicting and compressing data from irregularly located weather stations, or predicting behavior of customers of a mobile service provider. The construction of a meaningful graph plays a crucial role in the success of many graph-based data representations and algorithms, especially in the emerging field of signal processing on graphs. However, a meaningful graph is not always readily available from the data, nor easy to define depending on the application domain. In this paper, we address the problem of graph learning, where we are interested in learning graph topologies, namely, the relationships between data entities, that well explain the signal observations. In particular, we want to infer a graph such that the input data forms graph signals with smooth variations on the resulting topology. To this end, we adopt a factor analysis model for the graph signals and impose a Gaussian probabilistic prior on the latent variables that control these graph signals. We show that the Gaussian prior leads to an efficient representation that favors the smoothness property of the graph signals. We then propose an algorithm for learning graphs that enforce such smoothness property for the signal observations by minimizing the variations of the signals on the learned graph. Experiments on both synthetic and real world data demonstrate that the proposed graph learning framework can efficiently infer meaningful graph topologies from only the signal observations.