24,119 research outputs found
Distributed optimization over time-varying directed graphs
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
, 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
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
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
The sub-volume scaling of the entanglement entropy with the system's size,
, 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 in physically reasonable
models such as critical systems.
In this paper, we generalize the spin model of Bravyi et al [3] to all
integer spin- 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 , where
using the theory of Brownian excursions, we prove . 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 and that
the half-chain entanglement entropy to the leading order scales as
(Eq. 16). Geometrically, the ground state is seen as a uniform superposition of
all 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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