14,776 research outputs found
On Unconstrained Quasi-Submodular Function Optimization
With the extensive application of submodularity, its generalizations are
constantly being proposed. However, most of them are tailored for special
problems. In this paper, we focus on quasi-submodularity, a universal
generalization, which satisfies weaker properties than submodularity but still
enjoys favorable performance in optimization. Similar to the diminishing return
property of submodularity, we first define a corresponding property called the
{\em single sub-crossing}, then we propose two algorithms for unconstrained
quasi-submodular function minimization and maximization, respectively. The
proposed algorithms return the reduced lattices in iterations,
and guarantee the objective function values are strictly monotonically
increased or decreased after each iteration. Moreover, any local and global
optima are definitely contained in the reduced lattices. Experimental results
verify the effectiveness and efficiency of the proposed algorithms on lattice
reduction.Comment: 11 page
Simplifying Algebra in Feynman Graphs, Part III: Massive Vectors
A T-dualized selfdual inspired formulation of massive vector fields coupled
to arbitrary matter is generated; subsequently its perturbative series modeling
a spontaneously broken gauge theory is analyzed. The new Feynman rules and
external line factors are chirally minimized in the sense that only one type of
spin index occurs in the rules. Several processes are examined in detail and
the cross-sections formulated in this approach. A double line formulation of
the Lorentz algebra for Feynman diagrams is produced in this formalism, similar
to color ordering, which follows from a spin ordering of the Feynman rules. The
new double line formalism leads to further minimization of gauge invariant
scattering in perturbation theory. The dualized electroweak model is also
generated.Comment: 39 pages, LaTeX, 8 figure
Convex Relaxations for Permutation Problems
Seriation seeks to reconstruct a linear order between variables using
unsorted, pairwise similarity information. It has direct applications in
archeology and shotgun gene sequencing for example. We write seriation as an
optimization problem by proving the equivalence between the seriation and
combinatorial 2-SUM problems on similarity matrices (2-SUM is a quadratic
minimization problem over permutations). The seriation problem can be solved
exactly by a spectral algorithm in the noiseless case and we derive several
convex relaxations for 2-SUM to improve the robustness of seriation solutions
in noisy settings. These convex relaxations also allow us to impose structural
constraints on the solution, hence solve semi-supervised seriation problems. We
derive new approximation bounds for some of these relaxations and present
numerical experiments on archeological data, Markov chains and DNA assembly
from shotgun gene sequencing data.Comment: Final journal version, a few typos and references fixe
Reduced Complexity Filtering with Stochastic Dominance Bounds: A Convex Optimization Approach
This paper uses stochastic dominance principles to construct upper and lower
sample path bounds for Hidden Markov Model (HMM) filters. Given a HMM, by using
convex optimization methods for nuclear norm minimization with copositive
constraints, we construct low rank stochastic marices so that the optimal
filters using these matrices provably lower and upper bound (with respect to a
partially ordered set) the true filtered distribution at each time instant.
Since these matrices are low rank (say R), the computational cost of evaluating
the filtering bounds is O(XR) instead of O(X2). A Monte-Carlo importance
sampling filter is presented that exploits these upper and lower bounds to
estimate the optimal posterior. Finally, using the Dobrushin coefficient,
explicit bounds are given on the variational norm between the true posterior
and the upper and lower bounds
Ordered phases of XXZ-symmetric spin-1/2 zigzag ladder
Using bosonization approach, we derive an effective low-energy theory for
XXZ-symmetric spin-1/2 zigzag ladders and discuss its phase diagram by a
variational approach. A spin nematic phase emerges in a wide part of the phase
diagram, either critical or massive. Possible crossovers between the
spontaneously dimerized and spin nematic phases are discussed, and the
topological excitations in all phases identified.Comment: 14 pages, 3 figures. submitted to The European Physical Journal
Ground States of Fermionic lattice Hamiltonians with Permutation Symmetry
We study the ground states of lattice Hamiltonians that are invariant under
permutations, in the limit where the number of lattice sites, N -> \infty. For
spin systems, these are product states, a fact that follows directly from the
quantum de Finetti theorem. For fermionic systems, however, the problem is very
different, since mode operators acting on different sites do not commute, but
anti-commute. We construct a family of fermionic states, \cal{F}, from which
such ground states can be easily computed. They are characterized by few
parameters whose number only depends on M, the number of modes per lattice
site. We also give an explicit construction for M=1,2. In the first case,
\cal{F} is contained in the set of Gaussian states, whereas in the second it is
not. Inspired by that constructions, we build a set of fermionic variational
wave functions, and apply it to the Fermi-Hubbard model in two spatial
dimensions, obtaining results that go beyond the generalized Hartree-Fock
theory.Comment: 23 pages, published versio
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