135,776 research outputs found
The degree of the central curve in semidefinite, linear, and quadratic programming
The Zariski closure of the central path which interior point algorithms track
in convex optimization problems such as linear, quadratic, and semidefinite
programs is an algebraic curve. The degree of this curve has been studied in
relation to the complexity of these interior point algorithms, and for linear
programs it was computed by De Loera, Sturmfels, and Vinzant in 2012. We show
that the degree of the central curve for generic semidefinite programs is equal
to the maximum likelihood degree of linear concentration models. New results
from the intersection theory of the space of complete quadrics imply that this
is a polynomial in the size of semidefinite matrices with degree equal to the
number of constraints. Besides its degree we explore the arithmetic genus of
the same curve. We also compute the degree of the central curve for generic
linear programs with different techniques which extend to the computation of
the same degree for generic quadratic programs.Comment: 15 page
On the Curvature of the Central Path of Linear Programming Theory
We prove a linear bound on the average total curvature of the central path of
linear programming theory in terms on the number of independent variables of
the primal problem, and independent on the number of constraints.Comment: 24 pages. This is a fully revised version, and the last section of
the paper was rewritten, for clarit
Carving Out the Space of 4D CFTs
We introduce a new numerical algorithm based on semidefinite programming to
efficiently compute bounds on operator dimensions, central charges, and OPE
coefficients in 4D conformal and N=1 superconformal field theories. Using our
algorithm, we dramatically improve previous bounds on a number of CFT
quantities, particularly for theories with global symmetries. In the case of
SO(4) or SU(2) symmetry, our bounds severely constrain models of conformal
technicolor. In N=1 superconformal theories, we place strong bounds on
dim(Phi*Phi), where Phi is a chiral operator. These bounds asymptote to the
line dim(Phi*Phi) <= 2 dim(Phi) near dim(Phi) ~ 1, forbidding positive
anomalous dimensions in this region. We also place novel upper and lower bounds
on OPE coefficients of protected operators in the Phi x Phi OPE. Finally, we
find examples of lower bounds on central charges and flavor current two-point
functions that scale with the size of global symmetry representations. In the
case of N=1 theories with an SU(N) flavor symmetry, our bounds on current
two-point functions lie within an O(1) factor of the values realized in
supersymmetric QCD in the conformal window.Comment: 60 pages, 22 figure
Exploratory Analysis of Functional Data via Clustering and Optimal Segmentation
We propose in this paper an exploratory analysis algorithm for functional
data. The method partitions a set of functions into clusters and represents
each cluster by a simple prototype (e.g., piecewise constant). The total number
of segments in the prototypes, , is chosen by the user and optimally
distributed among the clusters via two dynamic programming algorithms. The
practical relevance of the method is shown on two real world datasets
Algebraic Boundaries of Convex Semi-algebraic Sets
We study the algebraic boundary of a convex semi-algebraic set via duality in
convex and algebraic geometry. We generalize the correspondence of facets of a
polytope to the vertices of the dual polytope to general semi-algebraic convex
bodies. In the general setup, exceptional families of extreme points might
exist and we characterize them semi-algebraically. We also give an algorithm to
compute a complete list of exceptional families, given the algebraic boundary
of the dual convex set.Comment: 13 pages, 2 figures; Comments welcom
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