Sampling of graph signals via randomized local aggregations

2015

A new scheme to sample signals defined in the nodes of a graph is proposed. The underlying assumption is that such signals admit a sparse representation in a frequency domain related to the structure of the graph, which is captured by the so-called graph-shift operator. Most of the works that have looked at this problem have focused on using the value of the signal observed at a subset of nodes to recover the signal in the entire graph. Differently, the sampling scheme proposed here uses as input observations taken at a single node. The observations correspond to sequential applications of the graph-shift operator, which are linear combinations of the information gathered by the neighbors of the node. When the graph corresponds to a directed cycle (which is the support of time-varying signals), our method is equivalent to the classical sampling in the time domain. When the graph is more general, we show that the Vandermonde structure of the sampling matrix, which is critical to guar...

IEEE Signal Processing Letters, 2019

We analyze the sampling and posterior recovery of diffused sparse graph signals from observations gathered at a single node using an aggregation sampling scheme. Diffused sparse graph signals can be modeled as the output of a linear graph filter to a sparse input, and are useful in scenarios where a few seeding (source) nodes generate a non-zero input which is then diffused according to the network dynamics dictated by the filter. Instead of considering a traditional setup where the observations correspond to the signal values at a subset of nodes, here the observations are obtained locally at a single node via the successive aggregation of its own value and that of its neighbors. Depending of the particular application, the goal is to use the local observations to recover the diffused signal or (the location and values of) the seeds. Different sampling configurations are investigated, including those of known and unknown location of the sources as well as that of the diffusing filter being unknown.

2015 International Conference on Sampling Theory and Applications (SampTA), 2015

We study signal recovery on graphs based on two sampling strategies: random sampling and experimentally designed sampling. We propose a new class of smooth graph signals, called approximately bandlimited. We then propose two recovery strategies based on random sampling and experimentally designed sampling. The proposed recovery strategy based on experimentally designed sampling uses sampling scores, which is similar to the leverage scores used in the matrix approximation. We show that while both strategies are unbiased estimators for the low-frequency components, the convergence rate of experimentally designed sampling is much faster than that of random sampling when a graph is irregular 1. We validate the proposed recovery strategies on three specific graphs: a ring graph, an Erdős-Rényi graph, and a star graph. The simulation results support the theoretical analysis.

2015 IEEE 6th International Workshop on Computational Advances in Multi-Sensor Adaptive Processing (CAMSAP), 2015

A scheme to sample bandlimited graph signals in the presence of noise is analyzed. Samples are aggregated at a single node by successive applications of the so-called graph-shift operator that encodes the local structure of the underlying graph. In contrast to the noiseless case, when noise is present the choice of the sampling node and the local sample-selection scheme play a major role in determining the interpolation error. We provide optimal sampling schemes for particular noise models. We also analyze and provide identifiability conditions for the case where the frequency support of the bandlimited signal is unknown. Finally, simulations with synthetic and real-world graph signals are used to illustrate the behavior of aggregation sampling in noisy scenarios.

IEEE Transactions on Signal and Information Processing over Networks

This paper builds theoretical foundations for the recovery of a newly proposed class of smooth graph signals, approximately bandlimited graph signals, under three sampling strategies: uniform sampling, experimentally designed sampling and active sampling. We then state minimax lower bounds on the maximum risk for the approximately bandlimited class under these three sampling strategies and show that active sampling cannot fundamentally outperform experimentally designed sampling. We propose a recovery strategy to compare uniform sampling with experimentally designed sampling. As the proposed recovery strategy lends itself well to statistical analysis, we derive the exact mean square error for each sampling strategy. To study convergence rates, we introduce two types of graphs and find that (1) the proposed recovery strategy achieves the optimal rates; and (2) the experimentally designed sampling fundamentally outperforms uniform sampling for Type-2 class of graphs. To validate our proposed recovery strategy, we test it on five specific graphs: a ring graph with k nearest neighbors, an Erdős-Rényi graph, a random geometric graph, a small-world graph and a power-law graph and find that experimental results match the proposed theory well. This work also presents a comprehensive explanation for when and why sampling for semi-supervised learning with graphs works.

arXiv: Signal Processing, 2018

We study the problem of sampling and reconstructing bandlimited graph signals where the objective is to select a subset of nodes of pre-specified cardinality that ensures interpolation of the original signal with the lowest possible reconstruction error. First, we consider a non-Bayesian scenario and propose an efficient iterative sampling procedure that in the noiseless case enables exact recovery of the original signal from the set of selected nodes. In the case of noisy measurements, a bound on the reconstruction error of the proposed algorithm is established. Then, we consider the Bayesian scenario where we formulate the sampling task as the problem of maximizing a monotone weak submodular function, and propose a randomized-greedy algorithm to find a sub-optimal subset. We derive a worst-case performance guarantee on the mean-square error achieved by the randomized-greedy algorithm for general non-stationary graph signals. The efficacy of the proposed methods is illustrated thro...

2015

We propose a sampling theory for signals that are supported on either directed or undirected graphs. The theory follows the same paradigm as classical sampling theory. We show that the perfect recovery is possible for graph signals bandlimited under the graph Fourier transform, and the sampled signal coefficients form a new graph signal, whose corresponding graph structure is constructed from the original graph structure, preserving frequency contents. By imposing a specific structure on the graph, graph signals reduce to finite discrete-time signals and the proposed sampling theory works reduces to classical signal processing. We further establish the connection to frames with maximal robustness to erasures as well as compressed sensing, and show how to choose the optimal sampling operator, how random sampling works on circulant graphs and Erdős-Rényi graphs, and how to handle full-band graph signals by using graph filter

Cooperative and Graph Signal Processing, 2018

The aim of this chapter is to give an overview of the recent advances related to sampling and recovery of signals defined over graphs. First, we illustrate the conditions for perfect recovery of bandlimited graph signals from samples collected over a selected set of vertexes. Then, we describe some sampling design criteria proposed in the literature to mitigate the effect of noise and model mismatching when performing graph signal recovery. Finally, we illustrate algorithms and optimal sampling strategies for adaptive recovery and tracking of dynamic graph signals, where both sampling set and signal values are allowed to vary with time. Numerical simulations carried out over both synthetic and real data illustrate the potential advantages of graph signal processing methods for sampling, interpolation, and tracking of signals observed over irregular domains such as, e.g., technological or biological networks.

ArXiv, 2021

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, that depends on the interplay between the dynamics over the graphs and sampling probability distributions. In optimal scenarios, no more than O(k log(k)) space-time samples are sufficient to ensure accurate and stable recovery of all k-bandlimited signals. In any case, dynamical sampling typically requires much fewer spatial samples than the static case by leveraging th...

2020

With the explosive growth of information and communication, data is being generated at an unprecedented rate from various sources, including multimedia, sensor networks, biological systems, social networks, and physical infrastructure. Research in graph signal processing aims to develop tools for processing such data by providing a framework for the analysis of high-dimensional datadefined on irregular graph domains. Graph signal processing extends fundamental signal processingconcepts to data supported on graphs that we refer to as graph signals. In this work, we study two fraternal problems: (1) sampling and (2) reconstruction of signals on graphs. Both of these problems are eminent topics in the field of signal processing over the past decades and have meaningful implications for many real-world problems including semi-supervised learning and activelearning on graphs. Sampling is the task of choosing or measuring some representative subset of the signal such that we can interpola...