256 research outputs found
A lower bound for the determinantal complexity of a hypersurface
We prove that the determinantal complexity of a hypersurface of degree is bounded below by one more than the codimension of the singular locus,
provided that this codimension is at least . As a result, we obtain that the
determinantal complexity of the permanent is . We also prove
that for , there is no nonsingular hypersurface in of
degree that has an expression as a determinant of a matrix of
linear forms while on the other hand for , a general determinantal
expression is nonsingular. Finally, we answer a question of Ressayre by showing
that the determinantal complexity of the unique (singular) cubic surface
containing a single line is .Comment: 7 pages, 0 figure
Polar Varieties and Efficient Real Elimination
Let be a smooth and compact real variety given by a reduced regular
sequence of polynomials . This paper is devoted to the
algorithmic problem of finding {\em efficiently} a representative point for
each connected component of . For this purpose we exhibit explicit
polynomial equations that describe the generic polar varieties of . This
leads to a procedure which solves our algorithmic problem in time that is
polynomial in the (extrinsic) description length of the input equations and in a suitably introduced, intrinsic geometric parameter, called
the {\em degree} of the real interpretation of the given equation system .Comment: 32 page
Using Elimination Theory to construct Rigid Matrices
The rigidity of a matrix A for target rank r is the minimum number of entries
of A that must be changed to ensure that the rank of the altered matrix is at
most r. Since its introduction by Valiant (1977), rigidity and similar
rank-robustness functions of matrices have found numerous applications in
circuit complexity, communication complexity, and learning complexity. Almost
all nxn matrices over an infinite field have a rigidity of (n-r)^2. It is a
long-standing open question to construct infinite families of explicit matrices
even with superlinear rigidity when r = Omega(n).
In this paper, we construct an infinite family of complex matrices with the
largest possible, i.e., (n-r)^2, rigidity. The entries of an n x n matrix in
this family are distinct primitive roots of unity of orders roughly exp(n^2 log
n). To the best of our knowledge, this is the first family of concrete (but not
entirely explicit) matrices having maximal rigidity and a succinct algebraic
description.
Our construction is based on elimination theory of polynomial ideals. In
particular, we use results on the existence of polynomials in elimination
ideals with effective degree upper bounds (effective Nullstellensatz). Using
elementary algebraic geometry, we prove that the dimension of the affine
variety of matrices of rigidity at most k is exactly n^2-(n-r)^2+k. Finally, we
use elimination theory to examine whether the rigidity function is
semi-continuous.Comment: 25 Pages, minor typos correcte
Semidefinite Representation of the -Ellipse
The -ellipse is the plane algebraic curve consisting of all points whose
sum of distances from given points is a fixed number. The polynomial
equation defining the -ellipse has degree if is odd and degree
if is even. We express this polynomial equation as
the determinant of a symmetric matrix of linear polynomials. Our representation
extends to weighted -ellipses and -ellipsoids in arbitrary dimensions,
and it leads to new geometric applications of semidefinite programming.Comment: 16 pages, 5 figure
Maximum Likelihood for Matrices with Rank Constraints
Maximum likelihood estimation is a fundamental optimization problem in
statistics. We study this problem on manifolds of matrices with bounded rank.
These represent mixtures of distributions of two independent discrete random
variables. We determine the maximum likelihood degree for a range of
determinantal varieties, and we apply numerical algebraic geometry to compute
all critical points of their likelihood functions. This led to the discovery of
maximum likelihood duality between matrices of complementary ranks, a result
proved subsequently by Draisma and Rodriguez.Comment: 22 pages, 1 figur
Maximum likelihood geometry in the presence of data zeros
Given a statistical model, the maximum likelihood degree is the number of
complex solutions to the likelihood equations for generic data. We consider
discrete algebraic statistical models and study the solutions to the likelihood
equations when the data contain zeros and are no longer generic. Focusing on
sampling and model zeros, we show that, in these cases, the solutions to the
likelihood equations are contained in a previously studied variety, the
likelihood correspondence. The number of these solutions give a lower bound on
the ML degree, and the problem of finding critical points to the likelihood
function can be partitioned into smaller and computationally easier problems
involving sampling and model zeros. We use this technique to compute a lower
bound on the ML degree for tensors of border
rank and tables of rank for ,
the first four values of for which the ML degree was previously unknown
A primal-dual formulation for certifiable computations in Schubert calculus
Formulating a Schubert problem as the solutions to a system of equations in
either Pl\"ucker space or in the local coordinates of a Schubert cell typically
involves more equations than variables. We present a novel primal-dual
formulation of any Schubert problem on a Grassmannian or flag manifold as a
system of bilinear equations with the same number of equations as variables.
This formulation enables numerical computations in the Schubert calculus to be
certified using algorithms based on Smale's \alpha-theory.Comment: 21 page
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