29 research outputs found
Explicit enumeration of triangulations with multiple boundaries
We enumerate rooted triangulations of a sphere with multiple holes by the
total number of edges and the length of each boundary component. The proof
relies on a combinatorial identity due to W.T. Tutte
Percolating paths through random points :
We prove consistency of four different approaches to formalizing the idea of
minimum average edge-length in a path linking some infinite subset of points of
a Poisson process. The approaches are (i) shortest path from origin through
some distinct points; (ii) shortest average edge-length in paths across the
diagonal of a large cube; (iii) shortest path through some specified proportion
of points in a large cube; (iv) translation-invariant measures on
paths in which contain a proportion of the Poisson points.
We develop basic properties of a normalized average length function
and pose challenging open problemComment: 28 page
On one property of distances in the infinite random quadrangulation
We show that the Schaeffer's tree for an infinite quadrangulation only
changes locally when changing the root of the quadrangulation. This follows
from one property of distances in the infinite uniform random quadrangulation
Stochastic Models for Phylogenetic Trees on Higher-order Taxa
Simple stochastic models for phylogenetic trees on species have been well
studied. But much paleontology data concerns time series or trees on
higher-order taxa, and any broad picture of relationships between extant groups
requires use of higher-order taxa. A coherent model for trees on (say) genera
should involve both a species-level model and a model for the classification
scheme by which species are assigned to genera. We present a general framework
for such models, and describe three alternate classification schemes. Combining
with the species-level model of Aldous-Popovic (2005), one gets models for
higher-order trees, and we initiate analytic study of such models. In
particular we derive formulas for the lifetime of genera, for the distribution
of number of species per genus, and for the offspring structure of the tree on
genera.Comment: 41 pages. Minor revision
Birth and death processes on certain random trees: Classification and stationary laws
The main substance of the paper concerns the growth rate and the
classification (ergodicity, transience) of a family of random trees. In the
basic model, new edges appear according to a Poisson process of parameter
and leaves can be deleted at a rate . The main results lay the
stress on the famous number . A complete classification of the process is
given in terms of the intensity factor : it is ergodic if
, and transient if . There is a phase transition
phenomenon: the usual region of null recurrence (in the parameter space) here
does not exist. This fact is rare for countable Markov chains with
exponentially distributed jumps. Some basic stationary laws are computed, e.g.
the number of vertices and the height. Various bounds, limit laws and
ergodic-like theorems are obtained, both for the transient and ergodic regimes.
In particular, when the system is transient, the height of the tree grows
linearly as the time , at a rate which is explicitly computed. Some
of the results are extended to the so-called multiclass model
Growth Rate and Ergodicity Conditions for a Class of Random Trees
Projet MEVALThis paper gives the growth rate and the ergodicity conditions for a simple class of random trees. New edges appear according to a Poisson process, and leaves can be deleted at a rate . The main results lay the stress on the famous number . In the case of a pure birth process, i.e. , the height of the tree at time grows linearly at the rate , in mean and almost surely as . When deletions of leaves are permitted, a complete classification of the process is given in terms of the intensity factor : it is ergodic if , and transient if . There is a phase transition phenomenon: the usual region of null recurrence (in the parameter space) here does not exist. This fact is rare for countable Markov chains with exponentially distributed jumps. Bounds are obtained for the transient regime
Reinforcement Learning based Curriculum Optimization for Neural Machine Translation
We consider the problem of making efficient use of heterogeneous training
data in neural machine translation (NMT). Specifically, given a training
dataset with a sentence-level feature such as noise, we seek an optimal
curriculum, or order for presenting examples to the system during training. Our
curriculum framework allows examples to appear an arbitrary number of times,
and thus generalizes data weighting, filtering, and fine-tuning schemes. Rather
than relying on prior knowledge to design a curriculum, we use reinforcement
learning to learn one automatically, jointly with the NMT system, in the course
of a single training run. We show that this approach can beat uniform and
filtering baselines on Paracrawl and WMT English-to-French datasets by up to
+3.4 BLEU, and match the performance of a hand-designed, state-of-the-art
curriculum.Comment: NAACL 2019 short paper. Reviewer comments not yet addresse
Birth and Death Processes on Certain Random Trees : Classification and Stationary Laws
Projet MEVALThe main substance of the paper concerns the growth rate and the classificatio- n (ergodicity, transience) of a family of random trees. In the basic model, new edges appear according to a Poisson process, and leaves can be deleted at a rate . The main results lay the stress on the famous number e. In the case of a pure birth process, i.e. =0, the height of the tree at time t grows linearly at the rate e, in mean and almost surely as t. When deletions of leaves are permitted, a complete classification of the process is given in terms of the intensity factor =/¸: it is ergodic if e^-1, and transient if >e^-1. There is a phase transition phenomenon: the usual region of null recurrence (in the parameter space) here does not exist. This fact is rare for countable Markov chains with exponentially distributed jump. Bounds are obtained for the transient regime. Some basic stationary laws are computed, e.g. the number of nodes and the height. An extension to the so-called multiclass case is presented, with a more complex classification