414 research outputs found
Membership in moment polytopes is in NP and coNP
We show that the problem of deciding membership in the moment polytope
associated with a finite-dimensional unitary representation of a compact,
connected Lie group is in NP and coNP. This is the first non-trivial result on
the computational complexity of this problem, which naively amounts to a
quadratically-constrained program. Our result applies in particular to the
Kronecker polytopes, and therefore to the problem of deciding positivity of the
stretched Kronecker coefficients. In contrast, it has recently been shown that
deciding positivity of a single Kronecker coefficient is NP-hard in general
[Ikenmeyer, Mulmuley and Walter, arXiv:1507.02955]. We discuss the consequences
of our work in the context of complexity theory and the quantum marginal
problem.Comment: 20 page
Efficient algorithms for tensor scaling, quantum marginals and moment polytopes
We present a polynomial time algorithm to approximately scale tensors of any
format to arbitrary prescribed marginals (whenever possible). This unifies and
generalizes a sequence of past works on matrix, operator and tensor scaling.
Our algorithm provides an efficient weak membership oracle for the associated
moment polytopes, an important family of implicitly-defined convex polytopes
with exponentially many facets and a wide range of applications. These include
the entanglement polytopes from quantum information theory (in particular, we
obtain an efficient solution to the notorious one-body quantum marginal
problem) and the Kronecker polytopes from representation theory (which capture
the asymptotic support of Kronecker coefficients). Our algorithm can be applied
to succinct descriptions of the input tensor whenever the marginals can be
efficiently computed, as in the important case of matrix product states or
tensor-train decompositions, widely used in computational physics and numerical
mathematics.
We strengthen and generalize the alternating minimization approach of
previous papers by introducing the theory of highest weight vectors from
representation theory into the numerical optimization framework. We show that
highest weight vectors are natural potential functions for scaling algorithms
and prove new bounds on their evaluations to obtain polynomial-time
convergence. Our techniques are general and we believe that they will be
instrumental to obtain efficient algorithms for moment polytopes beyond the
ones consider here, and more broadly, for other optimization problems
possessing natural symmetries
Polynomial-Time Amoeba Neighborhood Membership and Faster Localized Solving
We derive efficient algorithms for coarse approximation of algebraic
hypersurfaces, useful for estimating the distance between an input polynomial
zero set and a given query point. Our methods work best on sparse polynomials
of high degree (in any number of variables) but are nevertheless completely
general. The underlying ideas, which we take the time to describe in an
elementary way, come from tropical geometry. We thus reduce a hard algebraic
problem to high-precision linear optimization, proving new upper and lower
complexity estimates along the way.Comment: 15 pages, 9 figures. Submitted to a conference proceeding
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