17 research outputs found
Maximum information divergence from linear and toric models
We study the problem of maximizing information divergence from a new
perspective using logarithmic Voronoi polytopes. We show that for linear
models, the maximum is always achieved at the boundary of the probability
simplex. For toric models, we present an algorithm that combines the
combinatorics of the chamber complex with numerical algebraic geometry. We pay
special attention to reducible models and models of maximum likelihood degree
one.Comment: 33 pages, 6 figure
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
Average Behavior of Minimal Free Resolutions of Monomial Ideals
We describe the typical homological properties of monomial ideals defined by
random generating sets. We show that, under mild assumptions, random monomial
ideals (RMI's) will almost always have resolutions of maximal length; that is,
the projective dimension will almost always be , where is the number of
variables in the polynomial ring. We give a rigorous proof that
Cohen-Macaulayness is a "rare" property. We characterize when an RMI is
generic/strongly generic, and when it "is Scarf"---in other words, when the
algebraic Scarf complex of gives a minimal free
resolution of . As a result we see that, outside of a very specific ratio
of model parameters, RMI's are Scarf only when they are generic. We end with a
discussion of the average magnitude of Betti numbers.Comment: Final version, to appear in Proceedings of the AM
Symbolic powers of monomial ideals and Cohen-Macaulay vertex-weighted digraphs
In this paper we study irreducible representations and symbolic Rees algebras
of monomial ideals. Then we examine edge ideals associated to vertex-weighted
oriented graphs. These are digraphs having no oriented cycles of length two
with weights on the vertices. For a monomial ideal with no embedded primes we
classify the normality of its symbolic Rees algebra in terms of its primary
components. If the primary components of a monomial ideal are normal, we
present a simple procedure to compute its symbolic Rees algebra using Hilbert
bases, and give necessary and sufficient conditions for the equality between
its ordinary and symbolic powers. We give an effective characterization of the
Cohen--Macaulay vertex-weighted oriented forests. For edge ideals of transitive
weighted oriented graphs we show that Alexander duality holds. It is shown that
edge ideals of weighted acyclic tournaments are Cohen--Macaulay and satisfy
Alexander dualityComment: Special volume dedicated to Professor Antonio Campillo, Springer, to
appea
Logarithmic Voronoi Cells for Gaussian Models
We extend the theory of logarithmic Voronoi cells to Gaussian statistical
models. In general, a logarithmic Voronoi cell at a point on a Gaussian model
is a convex set contained in its log-normal spectrahedron. We show that for
models of ML degree one and linear covariance models the two sets coincide. In
particular, they are equal for both directed and undirected graphical models.
We introduce decomposition theory of logarithmic Voronoi cells for the latter
family. We also study covariance models, for which logarithmic Voronoi cells
are, in general, strictly contained in log-normal spectrahedra. We give an
explicit description of logarithmic Voronoi cells for the bivariate correlation
model and show that they are semi-algebraic sets. Finally, we state a
conjecture that logarithmic Voronoi cells for unrestricted correlation models
are not semi-algebraic.Comment: 24 page