Compressive sensing over graphs

Abstract

Abstract-In this paper, motivated by network inference and tomography applications, we study the problem of compressive sensing for sparse signal vectors over graphs. In particular, we are interested in recovering sparse vectors representing the properties of the edges from a graph. Unlike existing compressive sensing results, the collective additive measurements we are allowed to take must follow connected paths over the underlying graph. For a sufficiently connected graph with n nodes, it is shown that, using O(k log(n)) path measurements, we are able to recover any k-sparse link vector (with no more than k nonzero elements), even though the measurements have to follow the graph path constraints. We mainly show that the computationally efficient 1 minimization can provide theoretical guarantees for inferring such k-sparse vectors with O(k log(n)) path measurements from the graph.

A User's Guide to Compressed Sensing for Communications Systems

“…Since the increase in the number of paths, which corresponds to the number of measurements, results in the increase in the number of probe packets injected into the network, it is desirable to minimize the number of paths in order not to give unnecessary load to the network. As for the number of required measurements for the reconstruction of k-sparse vectors with random binary measurements matrices, it has been known that O(k log n k ) measurements are required [153], [154], while it has been shown that O(k log n) measurements are needed if the binary matrix is accompanied by a graph constraint [151], [155]. Moreover, a deterministic guarantee and a designed method of the routing matrix for the reconstruction of any 1-sparse signal are provided in [149], [150], taking advantage of the knowledge on compressed sensing using expander graphs [154], [156].…”

Quantized Network Coding for correlated sources

“…Consider tot (t) with the tail probability, as defined in (26), and an orthonormal transform matrix φ. Then, tot (t) = tot (t) · φ satisfies RIP of order k and constant δ k , with a probability exceeding,…”

“…Specifically, with the aid of the compressed sensing concepts, compression of inter-node correlated data without using their correlation model is done in [22,23]. Morevoer, in [26,27], theoretical discussion on sparse recovery of graph constrained measurements with an interest in network monitoring application is presented. Joint source, channel, and network coding was also proposed in [28], where random linear mixing was proposed for compression of temporally and spatially correlated sources.…”

Dynamic Network Delay Cartography

“…Recently, a compressive sampling-based approach has also been reported for predicting networkwide performance metrics [6], [7]. For instance, diffusion wavelets were utilized in [6] to obtain a compressible representation of the delays, and account for spatial and temporal correlations.…”