4,578 research outputs found
Facets of a mixed-integer bilinear covering set with bounds on variables
We derive a closed form description of the convex hull of mixed-integer
bilinear covering set with bounds on the integer variables. This convex hull
description is determined by considering some orthogonal disjunctive sets
defined in a certain way. This description does not introduce any new
variables, but consists of exponentially many inequalities. An extended
formulation with a few extra variables and much smaller number of constraints
is presented. We also derive a linear time separation algorithm for finding the
facet defining inequalities of this convex hull. We study the effectiveness of
the new inequalities and the extended formulation using some examples
Intersecting Faces: Non-negative Matrix Factorization With New Guarantees
Non-negative matrix factorization (NMF) is a natural model of admixture and
is widely used in science and engineering. A plethora of algorithms have been
developed to tackle NMF, but due to the non-convex nature of the problem, there
is little guarantee on how well these methods work. Recently a surge of
research have focused on a very restricted class of NMFs, called separable NMF,
where provably correct algorithms have been developed. In this paper, we
propose the notion of subset-separable NMF, which substantially generalizes the
property of separability. We show that subset-separability is a natural
necessary condition for the factorization to be unique or to have minimum
volume. We developed the Face-Intersect algorithm which provably and
efficiently solves subset-separable NMF under natural conditions, and we prove
that our algorithm is robust to small noise. We explored the performance of
Face-Intersect on simulations and discuss settings where it empirically
outperformed the state-of-art methods. Our work is a step towards finding
provably correct algorithms that solve large classes of NMF problems
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Zero-one IP problems: Polyhedral descriptions & cutting plane procedures
A systematic way for tightening an IP formulation is by employing classes of linear inequalities that define facets of the convex hull of the feasible integer points of the respective problems. Describing as well as identifying these inequalities will help in the efficiency of the LP-based cutting plane methods. In this report, we review classes of inequalities that partially described zero-one poly topes such as the 0-1 knapsack polytope, the set packing polytope and the travelling salesman polytope. Facets or valid inequalities derived from the 0-1 knapsack and the set packing polytopes are algorithmically identifie
Convex Hull Realizations of the Multiplihedra
We present a simple algorithm for determining the extremal points in
Euclidean space whose convex hull is the nth polytope in the sequence known as
the multiplihedra. This answers the open question of whether the multiplihedra
could be realized as convex polytopes. We use this realization to unite the
approach to A_n-maps of Iwase and Mimura to that of Boardman and Vogt. We
include a review of the appearance of the nth multiplihedron for various n in
the studies of higher homotopy commutativity, (weak) n-categories,
A_infinity-categories, deformation theory, and moduli spaces. We also include
suggestions for the use of our realizations in some of these areas as well as
in related studies, including enriched category theory and the graph
associahedra.Comment: typos fixed, introduction revise
Bounds on the Complexity of Halfspace Intersections when the Bounded Faces have Small Dimension
We study the combinatorial complexity of D-dimensional polyhedra defined as
the intersection of n halfspaces, with the property that the highest dimension
of any bounded face is much smaller than D. We show that, if d is the maximum
dimension of a bounded face, then the number of vertices of the polyhedron is
O(n^d) and the total number of bounded faces of the polyhedron is O(n^d^2). For
inputs in general position the number of bounded faces is O(n^d). For any fixed
d, we show how to compute the set of all vertices, how to determine the maximum
dimension of a bounded face of the polyhedron, and how to compute the set of
bounded faces in polynomial time, by solving a polynomial number of linear
programs
Generalized Bell Inequality Experiments and Computation
We consider general settings of Bell inequality experiments with many
parties, where each party chooses from a finite number of measurement settings
each with a finite number of outcomes. We investigate the constraints that Bell
inequalities place upon the correlations possible in a local hidden variable
theories using a geometrical picture of correlations. We show that local hidden
variable theories can be characterized in terms of limited computational
expressiveness, which allows us to characterize families of Bell inequalities.
The limited computational expressiveness for many settings (each with many
outcomes) generalizes previous results about the many-party situation each with
a choice of two possible measurements (each with two outcomes). Using this
computational picture we present generalizations of the Popescu-Rohrlich
non-local box for many parties and non-binary inputs and outputs at each site.
Finally, we comment on the effect of pre-processing on measurement data in our
generalized setting and show that it becomes problematic outside of the binary
setting, in that it allows local hidden variable theories to simulate maximally
non-local correlations such as those of these generalised Popescu-Rohrlich
non-local boxes.Comment: 16 pages, 2 figures, supplemental material available upon request.
Typos corrected and references adde
Noise Thresholds for Higher Dimensional Systems using the Discrete Wigner Function
For a quantum computer acting on d-dimensional systems, we analyze the
computational power of circuits wherein stabilizer operations are perfect and
we allow access to imperfect non-stabilizer states or operations. If the noise
rate affecting the non-stabilizer resource is sufficiently high, then these
states and operations can become simulable in the sense of the Gottesman-Knill
theorem, reducing the overall power of the circuit to no better than classical.
In this paper we find the depolarizing noise rate at which this happens, and
consequently the most robust non-stabilizer states and non-Clifford gates. In
doing so, we make use of the discrete Wigner function and derive facets of the
so-called qudit Clifford polytope i.e. the inequalities defining the convex
hull of all qudit Clifford gates. Our results for robust states are provably
optimal. For robust gates we find a critical noise rate that, as dimension
increases, rapidly approaches the the theoretical optimum of 100%. Some
connections with the question of qudit magic state distillation are discussed.Comment: 14 pages, 1 table; Minor changes vs. version
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