Passive and Active Sampling for Piecewise-Smooth Graph Signals Rohan Varma Dept. of Electrical and Computer Engineering Carnegie Mellon University Pittsburgh, U.S.A rohanv@andrew.cmu.edu Jelena Kovaˇ cevi´ c Tandon School of Engineering New York University New York City, U.S.A jelenak@nyu.edu Abstract—In this work, we study the sampling of piecewise smooth- graph signals that exhibit an inhomogeneous level of smoothness over the graph and are characterized by have abrupt, localized discontinuities between smooth regions of the graph. We propose an extension to the graph trend filtering framework under the sampling setting and present an ADMM-algorithm to efficiently reconstruct piecewise-smooth graph signals. Further, to alleviate the limitations of passive sampling in this setting, we develop an active sampling strategy that incorporates feedback to focus the sampling procedure near the boundary or discontinuities. We then conduct experiments that exhibit the performance of our algorithm on large complex graphs and validate the efficacy of our sampling strategies. Index Terms—sampling, piecewise smooth, graph signal processing, graph trend filtering I. I NTRODUCTION 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 [1]. Research in graph signal processing (GSP) aims to develop tools for processing such data by providing a framework for the analysis of high-dimensional data signals we refer to as graph signals defined on irregular graph domains [2], [3], [4]. Research on the sampling and recovery of graph signals has been prevalent in recent years [5], [6], [7], [8], [9]. The assumption that graph signals vary slowly or are smooth over the graph is a natural one to make. However, in social networks, within a given community or social circle, users’ profiles tend to be homogeneous, while within a different social circle they will be different, yet still homogeneous. Such signals are characterized by large variation between regions or pieces and slow variation within pieces. In this work, we study the sampling and reconstruction of such piecewise-smooth graph signals that exhibit a spatially inhomogeneous level of smoothness over regions of the graph and have abrupt, localized discontinuities. This class of piecewise-smooth signals is complementary to the class of smooth graph signals that exhibit spatially homogeneous levels of smoothness over the graph. The sampling of such smooth signals has been well studied in previous work both within the field of graph signal processing as well as in the context of Laplacian regularization. In the context of semi-supervised classification on graphs, each vertex represents one data point to which a label is associated and a graph can be formed by connecting vertices with weights corresponding to the affinity or distance between the data points in some feature space. It is then natural to assume that the label signal is piecewise-smooth on the graph. Since samples are often sparse or expensive, designing efficient sampling and reconstruction tools for This work is supported in part by NSF under grants CCF-1563918. semi-supervised classification and active learning is notably valuable. The graph trend filtering (GTF) framework [10], which applies total variation denoising on graphs [11], is a particularly flexible and attractive approach to process piecewise-smooth graph signals that is based on minimizing the ℓ1 norm of discrete graph differences. In this work, we present an extension to the GTF framework under the sampling setting, that is, where we only partially observe the signal. Most sampling strategies fall under the umbrellas of either (1) passive sampling where there is no feedback and we simply sample the space without any knowledge of key signal characteristics, or (2) active sampling where we can incorporate feedback in a sequential process. Unlike sampling smooth signals that have no discontinuities, the localized nature of the discontinuities in piecewise-smooth signals make the detection of these discontinuities inherently decoupled from the global or neighborhood features of the graph signal. It then follows that the passive sampling of piecewise-smooth graph signals is a significantly harder or even futile task than the same for globally smooth signals. For the latter, it is often sufficient to sample such that we uniformly cover the space. Consequently, we propose studying the active sampling of piecewise-smooth signals by designing algorithms and strategies that incorporate feedback. Particularly, we develop active sampling methods that can capitalize on the localized nature of the boundary by focusing the sampling process in the estimated vicinity of the boundary. II. GRAPH SIGNAL PROCESSING AND PIECEWISE-SMOOTH GRAPH SIGNALS A. Graphs We consider a weighted undirected graph G =(V , E , A), where V = {v1,...,vN } is the set of nodes, E = {e1,...,em} is the set of edges, and A =[A j,k ] ∈ R N×N is the graph shift operator [12], or the weighted adjacency matrix. The edge set E represents the connections of the undirected graph G, and the positive edge weight A j,k between nodes vj and v k measures the underlying relation between the j th and the kth node, such as a similarity, a dependency, or a communication pattern. Let a graph signal be defined as β =  β1,β2,...,βN  T ∈ R n , where βi denotes the signal coefficient at the ith node. Let Δ ∈ R m×N be the oriented incidence matrix of G, where each row corresponds to an edge. That is, if the edge ei =(j, k) ∈E connects the j th node to the kth node (j<k), the entries in the ith row of Δ is then given as Δ i,ℓ =    −  A j,k , ℓ = j ;  A j,k , ℓ = k; 0, otherwise .