289 research outputs found
Uniqueness of Nonnegative Tensor Approximations
We show that for a nonnegative tensor, a best nonnegative rank-r
approximation is almost always unique, its best rank-one approximation may
always be chosen to be a best nonnegative rank-one approximation, and that the
set of nonnegative tensors with non-unique best rank-one approximations form an
algebraic hypersurface. We show that the last part holds true more generally
for real tensors and thereby determine a polynomial equation so that a real or
nonnegative tensor which does not satisfy this equation is guaranteed to have a
unique best rank-one approximation. We also establish an analogue for real or
nonnegative symmetric tensors. In addition, we prove a singular vector variant
of the Perron--Frobenius Theorem for positive tensors and apply it to show that
a best nonnegative rank-r approximation of a positive tensor can never be
obtained by deflation. As an aside, we verify that the Euclidean distance (ED)
discriminants of the Segre variety and the Veronese variety are hypersurfaces
and give defining equations of these ED discriminants
The convex Positivstellensatz in a free algebra
Given a monic linear pencil L in g variables let D_L be its positivity
domain, i.e., the set of all g-tuples X of symmetric matrices of all sizes
making L(X) positive semidefinite. Because L is a monic linear pencil, D_L is
convex with interior, and conversely it is known that convex bounded
noncommutative semialgebraic sets with interior are all of the form D_L. The
main result of this paper establishes a perfect noncommutative
Nichtnegativstellensatz on a convex semialgebraic set. Namely, a noncommutative
polynomial p is positive semidefinite on D_L if and only if it has a weighted
sum of squares representation with optimal degree bounds: p = s^* s + \sum_j
f_j^* L f_j, where s, f_j are vectors of noncommutative polynomials of degree
no greater than 1/2 deg(p). This noncommutative result contrasts sharply with
the commutative setting, where there is no control on the degrees of s, f_j and
assuming only p nonnegative, as opposed to p strictly positive, yields a clean
Positivstellensatz so seldom that such cases are noteworthy.Comment: 22 page
New Dependencies of Hierarchies in Polynomial Optimization
We compare four key hierarchies for solving Constrained Polynomial
Optimization Problems (CPOP): Sum of Squares (SOS), Sum of Diagonally Dominant
Polynomials (SDSOS), Sum of Nonnegative Circuits (SONC), and the Sherali Adams
(SA) hierarchies. We prove a collection of dependencies among these hierarchies
both for general CPOPs and for optimization problems on the Boolean hypercube.
Key results include for the general case that the SONC and SOS hierarchy are
polynomially incomparable, while SDSOS is contained in SONC. A direct
consequence is the non-existence of a Putinar-like Positivstellensatz for
SDSOS. On the Boolean hypercube, we show as a main result that Schm\"udgen-like
versions of the hierarchies SDSOS*, SONC*, and SA* are polynomially equivalent.
Moreover, we show that SA* is contained in any Schm\"udgen-like hierarchy that
provides a O(n) degree bound.Comment: 26 pages, 4 figure
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