1,846 research outputs found

    Hitting time results for Maker-Breaker games

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    We study Maker-Breaker games played on the edge set of a random graph. Specifically, we consider the random graph process and analyze the first time in a typical random graph process that Maker starts having a winning strategy for his final graph to admit some property \mP. We focus on three natural properties for Maker's graph, namely being kk-vertex-connected, admitting a perfect matching, and being Hamiltonian. We prove the following optimal hitting time results: with high probability Maker wins the kk-vertex connectivity game exactly at the time the random graph process first reaches minimum degree 2k2k; with high probability Maker wins the perfect matching game exactly at the time the random graph process first reaches minimum degree 22; with high probability Maker wins the Hamiltonicity game exactly at the time the random graph process first reaches minimum degree 44. The latter two statements settle conjectures of Stojakovi\'{c} and Szab\'{o}.Comment: 24 page

    Expander 0\ell_0-Decoding

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    We introduce two new algorithms, Serial-0\ell_0 and Parallel-0\ell_0 for solving a large underdetermined linear system of equations y=AxRmy = Ax \in \mathbb{R}^m when it is known that xRnx \in \mathbb{R}^n has at most k<mk < m nonzero entries and that AA is the adjacency matrix of an unbalanced left dd-regular expander graph. The matrices in this class are sparse and allow a highly efficient implementation. A number of algorithms have been designed to work exclusively under this setting, composing the branch of combinatorial compressed-sensing (CCS). Serial-0\ell_0 and Parallel-0\ell_0 iteratively minimise yAx^0\|y - A\hat x\|_0 by successfully combining two desirable features of previous CCS algorithms: the information-preserving strategy of ER, and the parallel updating mechanism of SMP. We are able to link these elements and guarantee convergence in O(dnlogk)\mathcal{O}(dn \log k) operations by assuming that the signal is dissociated, meaning that all of the 2k2^k subset sums of the support of xx are pairwise different. However, we observe empirically that the signal need not be exactly dissociated in practice. Moreover, we observe Serial-0\ell_0 and Parallel-0\ell_0 to be able to solve large scale problems with a larger fraction of nonzeros than other algorithms when the number of measurements is substantially less than the signal length; in particular, they are able to reliably solve for a kk-sparse vector xRnx\in\mathbb{R}^n from mm expander measurements with n/m=103n/m=10^3 and k/mk/m up to four times greater than what is achievable by 1\ell_1-regularization from dense Gaussian measurements. Additionally, Serial-0\ell_0 and Parallel-0\ell_0 are observed to be able to solve large problems sizes in substantially less time than other algorithms for compressed sensing. In particular, Parallel-0\ell_0 is structured to take advantage of massively parallel architectures.Comment: 14 pages, 10 figure

    A robust parallel algorithm for combinatorial compressed sensing

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    In previous work two of the authors have shown that a vector xRnx \in \mathbb{R}^n with at most k<nk < n nonzeros can be recovered from an expander sketch AxAx in O(nnz(A)logk)\mathcal{O}(\mathrm{nnz}(A)\log k) operations via the Parallel-0\ell_0 decoding algorithm, where nnz(A)\mathrm{nnz}(A) denotes the number of nonzero entries in ARm×nA \in \mathbb{R}^{m \times n}. In this paper we present the Robust-0\ell_0 decoding algorithm, which robustifies Parallel-0\ell_0 when the sketch AxAx is corrupted by additive noise. This robustness is achieved by approximating the asymptotic posterior distribution of values in the sketch given its corrupted measurements. We provide analytic expressions that approximate these posteriors under the assumptions that the nonzero entries in the signal and the noise are drawn from continuous distributions. Numerical experiments presented show that Robust-0\ell_0 is superior to existing greedy and combinatorial compressed sensing algorithms in the presence of small to moderate signal-to-noise ratios in the setting of Gaussian signals and Gaussian additive noise
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