24,119 research outputs found

    Distributed optimization over time-varying directed graphs

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    We consider distributed optimization by a collection of nodes, each having access to its own convex function, whose collective goal is to minimize the sum of the functions. The communications between nodes are described by a time-varying sequence of directed graphs, which is uniformly strongly connected. For such communications, assuming that every node knows its out-degree, we develop a broadcast-based algorithm, termed the subgradient-push, which steers every node to an optimal value under a standard assumption of subgradient boundedness. The subgradient-push requires no knowledge of either the number of agents or the graph sequence to implement. Our analysis shows that the subgradient-push algorithm converges at a rate of O(ln(t)/t)O(\ln(t)/\sqrt{t}), where the constant depends on the initial values at the nodes, the subgradient norms, and, more interestingly, on both the consensus speed and the imbalances of influence among the nodes

    What majority decisions are possible with possible abstaining

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    Suppose we are given a family of choice functions on pairs from a given finite set. The set is considered as a set of alternatives (say candidates for an office) and the functions as potential "voters". The question is, what choice functions agree, on every pair, with the majority of some finite subfamily of the voters? For the problem as stated, a complete characterization was given in \citet{shelah2009mdp}, but here we allow each voter to abstain. There are four cases.Comment: 23 page

    Effective Generation of Subjectively Random Binary Sequences

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    We present an algorithm for effectively generating binary sequences which would be rated by people as highly likely to have been generated by a random process, such as flipping a fair coin.Comment: Introduction and Section 6 revise

    Power law violation of the area law in quantum spin chains

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    The sub-volume scaling of the entanglement entropy with the system's size, nn, has been a subject of vigorous study in the last decade [1]. The area law provably holds for gapped one dimensional systems [2] and it was believed to be violated by at most a factor of log(n)\log\left(n\right) in physically reasonable models such as critical systems. In this paper, we generalize the spin1-1 model of Bravyi et al [3] to all integer spin-ss chains, whereby we introduce a class of exactly solvable models that are physical and exhibit signatures of criticality, yet violate the area law by a power law. The proposed Hamiltonian is local and translationally invariant in the bulk. We prove that it is frustration free and has a unique ground state. Moreover, we prove that the energy gap scales as ncn^{-c}, where using the theory of Brownian excursions, we prove c2c\ge2. This rules out the possibility of these models being described by a conformal field theory. We analytically show that the Schmidt rank grows exponentially with nn and that the half-chain entanglement entropy to the leading order scales as n\sqrt{n} (Eq. 16). Geometrically, the ground state is seen as a uniform superposition of all ss-colored Motzkin walks. Lastly, we introduce an external field which allows us to remove the boundary terms yet retain the desired properties of the model. Our techniques for obtaining the asymptotic form of the entanglement entropy, the gap upper bound and the self-contained expositions of the combinatorial techniques, more akin to lattice paths, may be of independent interest.Comment: v3: 10+33 pages. In the PNAS publication, the abstract was rewritten and title changed to "Supercritical entanglement in local systems: Counterexample to the area law for quantum matter". The content is same otherwise. v2: a section was added with an external field to include a model with no boundary terms (open and closed chain). Asymptotic technique is improved. v1:37 pages, 10 figures. Proc. Natl. Acad. Sci. USA, (Nov. 2016
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