2,949 research outputs found
SOS-Hankel Tensors: Theory and Application
Hankel tensors arise from signal processing and some other applications. SOS
(sum-of-squares) tensors are positive semi-definite symmetric tensors, but not
vice versa. The problem for determining an even order symmetric tensor is an
SOS tensor or not is equivalent to solving a semi-infinite linear programming
problem, which can be done in polynomial time. On the other hand, the problem
for determining an even order symmetric tensor is positive semi-definite or not
is NP-hard. In this paper, we study SOS-Hankel tensors. Currently, there are
two known positive semi-definite Hankel tensor classes: even order complete
Hankel tensors and even order strong Hankel tensors. We show complete Hankel
tensors are strong Hankel tensors, and even order strong Hankel tensors are
SOS-Hankel tensors. We give several examples of positive semi-definite Hankel
tensors, which are not strong Hankel tensors. However, all of them are still
SOS-Hankel tensors. Does there exist a positive semi-definite non-SOS-Hankel
tensor? The answer to this question remains open. If the answer to this
question is no, then the problem for determining an even order Hankel tensor is
positive semi-definite or not is solvable in polynomial-time. An application of
SOS-Hankel tensors to the positive semi-definite tensor completion problem is
discussed. We present an ADMM algorithm for solving this problem. Some
preliminary numerical results on this algorithm are reported
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
Algorithms for Positive Semidefinite Factorization
This paper considers the problem of positive semidefinite factorization (PSD
factorization), a generalization of exact nonnegative matrix factorization.
Given an -by- nonnegative matrix and an integer , the PSD
factorization problem consists in finding, if possible, symmetric -by-
positive semidefinite matrices and such
that for , and . PSD
factorization is NP-hard. In this work, we introduce several local optimization
schemes to tackle this problem: a fast projected gradient method and two
algorithms based on the coordinate descent framework. The main application of
PSD factorization is the computation of semidefinite extensions, that is, the
representations of polyhedrons as projections of spectrahedra, for which the
matrix to be factorized is the slack matrix of the polyhedron. We compare the
performance of our algorithms on this class of problems. In particular, we
compute the PSD extensions of size for the
regular -gons when , and . We also show how to generalize our
algorithms to compute the square root rank (which is the size of the factors in
a PSD factorization where all factor matrices and have rank one)
and completely PSD factorizations (which is the special case where the input
matrix is symmetric and equality is required for all ).Comment: 21 pages, 3 figures, 3 table
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