655 research outputs found
Quasiflats in hierarchically hyperbolic spaces
The rank of a hierarchically hyperbolic space is the maximal number of
unbounded factors in a standard product region. For hierarchically hyperbolic
groups, this coincides with the maximal dimension of a quasiflat. Examples for
which the rank coincides with familiar quantities include: the dimension of
maximal Dehn twist flats for mapping class groups, the maximal rank of a free
abelian subgroup for right-angled Coxeter and Artin groups, and, for the
Weil--Petersson metric, the rank is the integer part of half the complex
dimension of Teichm\"{u}ller space.
We prove that any quasiflat of dimension equal to the rank lies within finite
distance of a union of standard orthants (under a mild condition satisfied by
all natural examples). This resolves outstanding conjectures when applied to
various examples. For mapping class group, we verify a conjecture of Farb; for
Teichm\"{u}ller space we answer a question of Brock; for CAT(0) cubical groups,
we handle special cases including right-angled Coxeter groups. An important
ingredient in the proof is that the hull of any finite set in an HHS is
quasi-isometric to a CAT(0) cube complex of dimension bounded by the rank.
We deduce a number of applications. For instance, we show that any
quasi-isometry between HHSs induces a quasi-isometry between certain simpler
HHSs. This allows one, for example, to distinguish quasi-isometry classes of
right-angled Artin/Coxeter groups. Another application is to quasi-isometric
rigidity. Our tools in many cases allow one to reduce the problem of
quasi-isometric rigidity for a given hierarchically hyperbolic group to a
combinatorial problem. We give a new proof of quasi-isometric rigidity of
mapping class groups, which, given our general quasiflats theorem, uses simpler
combinatorial arguments than in previous proofs.Comment: 58 pages, 6 figures. Revised according to referee comments. This is
the final pre-publication version; to appear in Duke Math. Jou
Polyhedral computational geometry for averaging metric phylogenetic trees
This paper investigates the computational geometry relevant to calculations
of the Frechet mean and variance for probability distributions on the
phylogenetic tree space of Billera, Holmes and Vogtmann, using the theory of
probability measures on spaces of nonpositive curvature developed by Sturm. We
show that the combinatorics of geodesics with a specified fixed endpoint in
tree space are determined by the location of the varying endpoint in a certain
polyhedral subdivision of tree space. The variance function associated to a
finite subset of tree space has a fixed algebraic formula within
each cell of the corresponding subdivision, and is continuously differentiable
in the interior of each orthant of tree space. We use this subdivision to
establish two iterative methods for producing sequences that converge to the
Frechet mean: one based on Sturm's Law of Large Numbers, and another based on
descent algorithms for finding optima of smooth functions on convex polyhedra.
We present properties and biological applications of Frechet means and extend
our main results to more general globally nonpositively curved spaces composed
of Euclidean orthants.Comment: 43 pages, 6 figures; v2: fixed typos, shortened Sections 1 and 5,
added counter example for polyhedrality of vistal subdivision in general
CAT(0) cubical complexes; v1: 43 pages, 5 figure
Boolean function monotonicity testing requires (almost) non-adaptive queries
We prove a lower bound of , for all , on the query
complexity of (two-sided error) non-adaptive algorithms for testing whether an
-variable Boolean function is monotone versus constant-far from monotone.
This improves a lower bound for the same problem that
was recently given in [CST14] and is very close to , which we
conjecture is the optimal lower bound for this model
Partially ample line bundles on toric varieties
In this note we study properties of partially ample line bundles on
simplicial projective toric varieties. We prove that the cone of q-ample line
bundles is a union of rational polyhedral cones, and calculate these cones in
examples. We prove a restriction theorem for big q-ample line bundles, and
deduce that q-ampleness of the anticanonical bundle is not invariant under
flips. Finally we prove a Kodaira-type vanishing theorem for q-ample line
bundles.Comment: 12 pages, 2 figures; v.2: proofs simplified, lots of material added,
new autho
When Does a Mixture of Products Contain a Product of Mixtures?
We derive relations between theoretical properties of restricted Boltzmann
machines (RBMs), popular machine learning models which form the building blocks
of deep learning models, and several natural notions from discrete mathematics
and convex geometry. We give implications and equivalences relating
RBM-representable probability distributions, perfectly reconstructible inputs,
Hamming modes, zonotopes and zonosets, point configurations in hyperplane
arrangements, linear threshold codes, and multi-covering numbers of hypercubes.
As a motivating application, we prove results on the relative representational
power of mixtures of product distributions and products of mixtures of pairs of
product distributions (RBMs) that formally justify widely held intuitions about
distributed representations. In particular, we show that a mixture of products
requiring an exponentially larger number of parameters is needed to represent
the probability distributions which can be obtained as products of mixtures.Comment: 32 pages, 6 figures, 2 table
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