29,852 research outputs found
Harmonic mappings valued in the Wasserstein space
We propose a definition of the Dirichlet energy (which is roughly speaking
the integral of the square of the gradient) for mappings mu : Omega -> (P(D),
W\_2) defined over a subset Omega of R^p and valued in the space P(D) of
probability measures on a compact convex subset D of R^q endowed with the
quadratic Wasserstein distance. Our definition relies on a straightforward
generalization of the Benamou-Brenier formula (already introduced by Brenier)
but is also equivalent to the definition of Koorevaar, Schoen and Jost as limit
of approximate Dirichlet energies, and to the definition of Reshetnyak of
Sobolev spaces valued in metric spaces. We study harmonic mappings, i.e.
minimizers of the Dirichlet energy provided that the values on the boundary d
Omega are fixed. The notion of constant-speed geodesics in the Wasserstein
space is recovered by taking for Omega a segment of R. As the Wasserstein space
(P(D), W\_2) is positively curved in the sense of Alexandrov we cannot apply
the theory of Koorevaar, Schoen and Jost and we use instead arguments based on
optimal transport. We manage to get existence of harmonic mappings provided
that the boundary values are Lipschitz on d Omega, uniqueness is an open
question. If Omega is a segment of R, it is known that a curve valued in the
Wasserstein space P(D) can be seen as a superposition of curves valued in D. We
show that it is no longer the case in higher dimensions: a generic mapping
Omega -> P(D) cannot be represented as the superposition of mappings Omega ->
D. We are able to show the validity of a maximum principle: the composition
F(mu) of a function F : P(D) -> R convex along generalized geodesics and a
harmonic mapping mu : Omega -> P(D) is a subharmonic real-valued function. We
also study the special case where we restrict ourselves to a given family of
elliptically contoured distributions (a finite-dimensional and geodesically
convex submanifold of (P(D), W\_2) which generalizes the case of Gaussian
measures) and show that it boils down to harmonic mappings valued in the
Riemannian manifold of symmetric matrices endowed with the distance coming from
optimal transport
Bayesian Bootstrap Analysis of Systems of Equations
Research Methods/ Statistical Methods,
Coverage, Continuity and Visual Cortical Architecture
The primary visual cortex of many mammals contains a continuous
representation of visual space, with a roughly repetitive aperiodic map of
orientation preferences superimposed. It was recently found that orientation
preference maps (OPMs) obey statistical laws which are apparently invariant
among species widely separated in eutherian evolution. Here, we examine whether
one of the most prominent models for the optimization of cortical maps, the
elastic net (EN) model, can reproduce this common design. The EN model
generates representations which optimally trade of stimulus space coverage and
map continuity. While this model has been used in numerous studies, no
analytical results about the precise layout of the predicted OPMs have been
obtained so far. We present a mathematical approach to analytically calculate
the cortical representations predicted by the EN model for the joint mapping of
stimulus position and orientation. We find that in all previously studied
regimes, predicted OPM layouts are perfectly periodic. An unbiased search
through the EN parameter space identifies a novel regime of aperiodic OPMs with
pinwheel densities lower than found in experiments. In an extreme limit,
aperiodic OPMs quantitatively resembling experimental observations emerge.
Stabilization of these layouts results from strong nonlocal interactions rather
than from a coverage-continuity-compromise. Our results demonstrate that
optimization models for stimulus representations dominated by nonlocal
suppressive interactions are in principle capable of correctly predicting the
common OPM design. They question that visual cortical feature representations
can be explained by a coverage-continuity-compromise.Comment: 100 pages, including an Appendix, 21 + 7 figure
Equivalence theory for density estimation, Poisson processes and Gaussian white noise with drift
This paper establishes the global asymptotic equivalence between a Poisson
process with variable intensity and white noise with drift under sharp
smoothness conditions on the unknown function. This equivalence is also
extended to density estimation models by Poissonization. The asymptotic
equivalences are established by constructing explicit equivalence mappings. The
impact of such asymptotic equivalence results is that an investigation in one
of these nonparametric models automatically yields asymptotically analogous
results in the other models.Comment: Published at http://dx.doi.org/10.1214/009053604000000012 in the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org
- âŠ