393 research outputs found
Random discrete concave functions on an equilateral lattice with periodic Hessians
Motivated by connections to random matrices, Littlewood-Richardson
coefficients and tilings, we study random discrete concave functions on an
equilateral lattice. We show that such functions having a periodic Hessian of a
fixed average value concentrate around a quadratic function. We consider
the set of all concave functions on an equilateral lattice that
when shifted by an element of have a periodic discrete Hessian,
with period . We add a convex quadratic of Hessian ; the sum is
then periodic with period , and view this as a mean zero function
on the set of vertices of a torus whose
Hessian is dominated by . The resulting set of semiconcave functions forms a
convex polytope . The diameter of is bounded
below by , where is a positive constant depending only on .
Our main result is that under certain conditions, that are met for example when
, for any we have if
is sampled from the uniform measure on . Each
corresponds to a kind of honeycomb. We obtain concentration results for these
as well.Comment: 56 pages. arXiv admin note: substantial text overlap with
arXiv:1909.0858
Tropical Theta Functions and Log Calabi-Yau Surfaces
We generalize the standard combinatorial techniques of toric geometry to the
study of log Calabi-Yau surfaces. The character and cocharacter lattices are
replaced by certain integral linear manifolds described by Gross, Hacking, and
Keel, and monomials on toric varieties are replaced with the canonical theta
functions which GHK defined using ideas from mirror symmetry. We describe the
tropicalizations of theta functions and use them to generalize the dual pairing
between the character and cocharacter lattices. We use this to describe
generalizations of dual cones, Newton and polar polytopes, Minkowski sums, and
finite Fourier series expansions. We hope that these techniques will generalize
to higher-rank cluster varieties.Comment: 40 pages, 2 figures. The final publication is available at Springer
via http://dx.doi.org/10.1007/s00029-015-0221-y, Selecta Math. (2016
Classification and Geometry of General Perceptual Manifolds
Perceptual manifolds arise when a neural population responds to an ensemble
of sensory signals associated with different physical features (e.g.,
orientation, pose, scale, location, and intensity) of the same perceptual
object. Object recognition and discrimination requires classifying the
manifolds in a manner that is insensitive to variability within a manifold. How
neuronal systems give rise to invariant object classification and recognition
is a fundamental problem in brain theory as well as in machine learning. Here
we study the ability of a readout network to classify objects from their
perceptual manifold representations. We develop a statistical mechanical theory
for the linear classification of manifolds with arbitrary geometry revealing a
remarkable relation to the mathematics of conic decomposition. Novel
geometrical measures of manifold radius and manifold dimension are introduced
which can explain the classification capacity for manifolds of various
geometries. The general theory is demonstrated on a number of representative
manifolds, including L2 ellipsoids prototypical of strictly convex manifolds,
L1 balls representing polytopes consisting of finite sample points, and
orientation manifolds which arise from neurons tuned to respond to a continuous
angle variable, such as object orientation. The effects of label sparsity on
the classification capacity of manifolds are elucidated, revealing a scaling
relation between label sparsity and manifold radius. Theoretical predictions
are corroborated by numerical simulations using recently developed algorithms
to compute maximum margin solutions for manifold dichotomies. Our theory and
its extensions provide a powerful and rich framework for applying statistical
mechanics of linear classification to data arising from neuronal responses to
object stimuli, as well as to artificial deep networks trained for object
recognition tasks.Comment: 24 pages, 12 figures, Supplementary Material
Solving nonconvex planar location problems by nite dominating sets
It is well-known that some of the classical location problems with polyhedral
gauges can be solved in polynomial time by nding a fi nite dominating set, i.e. a finite set of candidates guaranteed to contain at least one optimal location.
In this paper it is fi rst established that this result holds for a much larger
class of problems than currently considered in the literature. The model for
which this result can be proven includes, for instance, location problems with
attraction and repulsion, and location-allocation problems. Next, it is shown that the approximation of general gauges by polyhedral ones in the objective function of our general model can be analyzed with regard to the subsequent error in the optimal ob jective value. For the approximation
problem two di erent approaches are described, the sandwich procedure and the greedy algorithm. Both of these approaches lead - for fixed e - to polynomial approximation algorithms with accuracy for solving the
general model considered in this paper.Dirección General de Enseñanza Superio
Minimizing the average distance to a closest leaf in a phylogenetic tree
When performing an analysis on a collection of molecular sequences, it can be
convenient to reduce the number of sequences under consideration while
maintaining some characteristic of a larger collection of sequences. For
example, one may wish to select a subset of high-quality sequences that
represent the diversity of a larger collection of sequences. One may also wish
to specialize a large database of characterized "reference sequences" to a
smaller subset that is as close as possible on average to a collection of
"query sequences" of interest. Such a representative subset can be useful
whenever one wishes to find a set of reference sequences that is appropriate to
use for comparative analysis of environmentally-derived sequences, such as for
selecting "reference tree" sequences for phylogenetic placement of metagenomic
reads. In this paper we formalize these problems in terms of the minimization
of the Average Distance to the Closest Leaf (ADCL) and investigate algorithms
to perform the relevant minimization. We show that the greedy algorithm is not
effective, show that a variant of the Partitioning Among Medoids (PAM)
heuristic gets stuck in local minima, and develop an exact dynamic programming
approach. Using this exact program we note that the performance of PAM appears
to be good for simulated trees, and is faster than the exact algorithm for
small trees. On the other hand, the exact program gives solutions for all
numbers of leaves less than or equal to the given desired number of leaves,
while PAM only gives a solution for the pre-specified number of leaves. Via
application to real data, we show that the ADCL criterion chooses chimeric
sequences less often than random subsets, while the maximization of
phylogenetic diversity chooses them more often than random. These algorithms
have been implemented in publicly available software.Comment: Please contact us with any comments or questions
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