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The spectral theory of graphs provides a bridge between classical signal processing and the nascent field of graph signal processing. In this paper, a spectral graph analogy to Heisenberg's celebrated uncertainty principle is developed. Just as the classical result provides a tradeoff between signal localization in time and frequency, this result provides a(More)
The Kaczmarz method, or the algebraic reconstruction technique (ART), is a popular method for solving large-scale overdetermined systems of equations. Recently, Strohmer et al. proposed the ran-domized Kaczmarz algorithm, an improvement that guarantees exponential convergence to the solution. This has spurred much interest in the algorithm and its(More)
The classical uncertainty principle provides a fundamental tradeoff in the localization of a signal in the time and frequency domains. In this paper we describe a similar tradeoff for signals defined on graphs. We describe the notions of " spread " in the graph and spectral domains, using the eigenvectors of the graph Laplacian as a surrogate Fourier basis.(More)
—Distributed detection of information flows spanning many nodes in a wireless sensor network is considered. In such a system, eavesdroppers are deployed near several nodes in the network. As data may be encrypted or padded, the eavesdroppers can only measure packet timestamps. Each eavesdropper, given a sequence of timestamps, must compress the information(More)
—The problem of detecting multi-hop information flows subject to communication constraints is considered. In a distributed detection scheme, eavesdroppers are deployed near nodes in a network, each able to measure the transmission times-tamps of a single node. The eavesdroppers must then compress the information and transmit it to a fusion center, which(More)
We consider the problem of distinguishing between two hypotheses: that a sequence of signals on a large graph consists entirely of noise, or that it contains a realization of a random walk buried in the noise. The problem of computing the error exponent of the optimal detector is simple to formulate, but exhibits deep connections to problems known to be(More)
—We provide a complete characterization of the randomized Kaczmarz algorithm (RKA) for inconsistent linear systems. The Kaczmarz algorithm, known in some fields as the algebraic reconstruction technique, is a classical method for solving large-scale overdetermined linear systems through a sequence of projection operators; the randomized Kaczmarz algorithm(More)
We study the problem of detecting a random walk on a graph from a sequence of noisy measurements at every node. There are two hypotheses: either every observation is just meaningless zero-mean Gaussian noise, or at each time step exactly one node has an elevated mean, with its location following a random walk on the graph over time. We want to exploit(More)