51,185 research outputs found
The geometric complexity of special Lagrangian -cones
We prove a number of results relating various measures (volume, Legendrian
index, stability index, and spectral curve genus) of the geometric complexity
of special Lagrangian -cones. We explain how these results fit into a
program to understand the "most common" three-dimensional isolated
singularities of special Lagrangian submanifolds in almost Calabi-Yau
manifolds.Comment: Revised version accepted for publication in Inventiones Mathematicae.
46 pages, 2 tables. Reference added relating to Theorem B. Section 3.4.2,
section 4.2 and Appendix B streamlined. Typographical errors corrected and
references update
Lower bounds on the size of semidefinite programming relaxations
We introduce a method for proving lower bounds on the efficacy of
semidefinite programming (SDP) relaxations for combinatorial problems. In
particular, we show that the cut, TSP, and stable set polytopes on -vertex
graphs are not the linear image of the feasible region of any SDP (i.e., any
spectrahedron) of dimension less than , for some constant .
This result yields the first super-polynomial lower bounds on the semidefinite
extension complexity of any explicit family of polytopes.
Our results follow from a general technique for proving lower bounds on the
positive semidefinite rank of a matrix. To this end, we establish a close
connection between arbitrary SDPs and those arising from the sum-of-squares SDP
hierarchy. For approximating maximum constraint satisfaction problems, we prove
that SDPs of polynomial-size are equivalent in power to those arising from
degree- sum-of-squares relaxations. This result implies, for instance,
that no family of polynomial-size SDP relaxations can achieve better than a
7/8-approximation for MAX-3-SAT
Lower Bounds on the Oracle Complexity of Nonsmooth Convex Optimization via Information Theory
We present an information-theoretic approach to lower bound the oracle
complexity of nonsmooth black box convex optimization, unifying previous lower
bounding techniques by identifying a combinatorial problem, namely string
guessing, as a single source of hardness. As a measure of complexity we use
distributional oracle complexity, which subsumes randomized oracle complexity
as well as worst-case oracle complexity. We obtain strong lower bounds on
distributional oracle complexity for the box , as well as for the
-ball for (for both low-scale and large-scale regimes),
matching worst-case upper bounds, and hence we close the gap between
distributional complexity, and in particular, randomized complexity, and
worst-case complexity. Furthermore, the bounds remain essentially the same for
high-probability and bounded-error oracle complexity, and even for combination
of the two, i.e., bounded-error high-probability oracle complexity. This
considerably extends the applicability of known bounds
Lifts of convex sets and cone factorizations
In this paper we address the basic geometric question of when a given convex
set is the image under a linear map of an affine slice of a given closed convex
cone. Such a representation or 'lift' of the convex set is especially useful if
the cone admits an efficient algorithm for linear optimization over its affine
slices. We show that the existence of a lift of a convex set to a cone is
equivalent to the existence of a factorization of an operator associated to the
set and its polar via elements in the cone and its dual. This generalizes a
theorem of Yannakakis that established a connection between polyhedral lifts of
a polytope and nonnegative factorizations of its slack matrix. Symmetric lifts
of convex sets can also be characterized similarly. When the cones live in a
family, our results lead to the definition of the rank of a convex set with
respect to this family. We present results about this rank in the context of
cones of positive semidefinite matrices. Our methods provide new tools for
understanding cone lifts of convex sets.Comment: 20 pages, 2 figure
Projective schemes: What is Computable in low degree?
This article first presents two examples of algorithms that extracts
information on scheme out of its defining equations. We also give a review on
the notion of Castelnuovo-Mumford regularity, its main properties (in
particular its relation to computational issues) and different ways that were
used to estimate it
Lower bounds on the entanglement needed to play XOR non-local games
We give an explicit family of XOR games with O(n)-bit questions requiring 2^n
ebits to play near-optimally. More generally we introduce a new technique for
proving lower bounds on the amount of entanglement required by an XOR game: we
show that near-optimal strategies for an XOR game G correspond to approximate
representations of a certain C^*-algebra associated to G. Our results extend an
earlier theorem of Tsirelson characterising the set of quantum strategies which
implement extremal quantum correlations.Comment: 20 pages, no figures. Corrected abstract, body of paper unchange
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