341 research outputs found
Exponential convergence for a convexifying equation and a non-autonomous gradient flow for global minimization
We consider an evolution equation similar to that introduced by Vese and
whose solution converges in large time to the convex envelope of the initial
datum. We give a stochastic control representation for the solution from which
we deduce, under quite general assumptions that the convergence in the
Lipschitz norm is in fact exponential in time. We then introduce a
non-autonomous gradient flow and prove that its trajectories all converge to
minimizers of the convex envelope
Vector Quantile Regression: An Optimal Transport Approach
We propose a notion of conditional vector quantile function and a vector
quantile regression. A \emph{conditional vector quantile function} (CVQF) of a
random vector , taking values in given covariates ,
taking values in , is a map ,
which is monotone, in the sense of being a gradient of a convex function, and
such that given that vector follows a reference non-atomic distribution
, for instance uniform distribution on a unit cube in , the
random vector has the distribution of conditional on
. Moreover, we have a strong representation, almost
surely, for some version of . The \emph{vector quantile regression} (VQR) is
a linear model for CVQF of given . Under correct specification, the
notion produces strong representation, , for
denoting a known set of transformations of , where is a monotone map, the gradient of a convex function, and
the quantile regression coefficients have the
interpretations analogous to that of the standard scalar quantile regression.
As becomes a richer class of transformations of , the model becomes
nonparametric, as in series modelling. A key property of VQR is the embedding
of the classical Monge-Kantorovich's optimal transportation problem at its core
as a special case. In the classical case, where is scalar, VQR reduces to a
version of the classical QR, and CVQF reduces to the scalar conditional
quantile function. An application to multiple Engel curve estimation is
considered
Pareto efficiency for the concave order and multivariate comonotonicity
In this paper, we focus on efficient risk-sharing rules for the concave
dominance order. For a univariate risk, it follows from a comonotone dominance
principle, due to Landsberger and Meilijson [25], that efficiency is
characterized by a comonotonicity condition. The goal of this paper is to
generalize the comonotone dominance principle as well as the equivalence
between efficiency and comonotonicity to the multi-dimensional case. The
multivariate setting is more involved (in particular because there is no
immediate extension of the notion of comonotonicity) and we address it using
techniques from convex duality and optimal transportation
Letter from Alfred Guillaume Gabriel, Count d\u27Orsay to an unidentified recipient
Letter from Alfred Guillaume Gabriel, Count d\u27Orsay to an unidentified individual, dated \u27Marcredi\u27.https://scholarworks.umt.edu/whicker/1021/thumbnail.jp
Lithograph of Thomas Carlyle by Alfred Guillaume Gabriel, Count d\u27Orsay
Lithograph of Thomas Carlyle by D\u27Orsay, dated May 1839.https://scholarworks.umt.edu/whicker/1022/thumbnail.jp
SISTA: learning optimal transport costs under sparsity constraints
In this paper, we describe a novel iterative procedure called SISTA to learn
the underlying cost in optimal transport problems. SISTA is a hybrid between
two classical methods, coordinate descent ("S"-inkhorn) and proximal gradient
descent ("ISTA"). It alternates between a phase of exact minimization over the
transport potentials and a phase of proximal gradient descent over the
parameters of the transport cost. We prove that this method converges linearly,
and we illustrate on simulated examples that it is significantly faster than
both coordinate descent and ISTA. We apply it to estimating a model of
migration, which predicts the flow of migrants using country-specific
characteristics and pairwise measures of dissimilarity between countries. This
application demonstrates the effectiveness of machine learning in quantitative
social sciences
Pareto efficiency for the concave order and multivariate comonotonicity
This paper studies efficient risk-sharing rules for the concave dominance order. For a univariate risk, it follows from a comonotone dominance principle, due to Landsberger and Meilijson (1994), that efficiency is characterized by a comonotonicity condition. The goal of the paper is to generalize the comonotone dominance principle as well as the equivalence between efficiency and comonotonicity to the multidimensional case. The multivariate case is more involved (in particular because there is no immediate extension of the notion of comonotonicity), and it is addressed by using techniques from convex duality and optimal transportation
A Spitzer Search for Water in the Transiting Exoplanet HD189733b
We present Spitzer Space Telescope observations of the extrasolar planet
HD189733b primary transit, obtained simultaneously at 3.6 and 5.8 microns with
the Infrared Array Camera. The system parameters, including planetary radius,
stellar radius, and impact parameter are derived from fits to the transit light
curves at both wavelengths. We measure two consistent planet-to-star radius
ratios, (Rp/Rs)[3.6m] = 0.1560 +/- 0.0008(stat) +/- 0.0002(syst) and
(Rp/Rs)[5.8m] = 0.1541 +/- 0.0009(stat) +/- 0.0009(syst), which include
both the random and systematic errors in the transit baseline. Although planet
radii are determined at 1%-accuracy, if all uncertainties are taken into
account the resulting error bars are still too large to allow for the detection
of atmospheric constituants like water vapour. This illustrates the need to
observe multiple transits with the longest possible out-of-transit baseline, in
order to achieve the precision required by transmission spectroscopy of giant
extrasolar planets.Comment: Accepted in The Astrophysical Journal Letter
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