138 research outputs found
Geometry and entropies in a fixed conformal class on surfaces
We show the flexibility of the metric entropy and obtain additional
restrictions on the topological entropy of geodesic flow on closed surfaces of
negative Euler characteristic with smooth non-positively curved Riemannian
metrics with fixed total area in a fixed conformal class. Moreover, we obtain a
collar lemma, a thick-thin decomposition, and precompactness for the considered
class of metrics. Also, we extend some of the results to metrics of fixed total
area in a fixed conformal class with no focal points and with some integral
bounds on the positive part of the Gaussian curvature.Comment: Minor changes to exposition. Final version, to appear in Annales de
l'Institut Fourie
Bounding errors of Expectation-Propagation
Expectation Propagation is a very popular algorithm for variational
inference, but comes with few theoretical guarantees. In this article, we prove
that the approximation errors made by EP can be bounded. Our bounds have an
asymptotic interpretation in the number of datapoints, which allows us to
study EP's convergence with respect to the true posterior. In particular, we
show that EP converges at a rate of for the mean, up to
an order of magnitude faster than the traditional Gaussian approximation at the
mode. We also give similar asymptotic expansions for moments of order 2 to 4,
as well as excess Kullback-Leibler cost (defined as the additional KL cost
incurred by using EP rather than the ideal Gaussian approximation). All these
expansions highlight the superior convergence properties of EP. Our approach
for deriving those results is likely applicable to many similar approximate
inference methods. In addition, we introduce bounds on the moments of
log-concave distributions that may be of independent interest.Comment: Accepted and published at NIPS 201
Knot theory of R-covered Anosov flows: homotopy versus isotopy of closed orbits
In this article, we study the knots realized by periodic orbits of R-covered
Anosov flows in compact 3-manifolds. We show that if two orbits are freely
homotopic then in fact they are isotopic. We show that lifts of periodic orbits
to the universal cover are unknotted. When the manifold is atoroidal, we deduce
some finer properties regarding the existence of embedded cylinders connecting
two given homotopic orbits.Comment: 20 pages, 9 figure
The Poisson transform for unnormalised statistical models
Contrary to standard statistical models, unnormalised statistical models only
specify the likelihood function up to a constant. While such models are natural
and popular, the lack of normalisation makes inference much more difficult.
Here we show that inferring the parameters of a unnormalised model on a space
can be mapped onto an equivalent problem of estimating the intensity
of a Poisson point process on . The unnormalised statistical model now
specifies an intensity function that does not need to be normalised.
Effectively, the normalisation constant may now be inferred as just another
parameter, at no loss of information. The result can be extended to cover
non-IID models, which includes for example unnormalised models for sequences of
graphs (dynamical graphs), or for sequences of binary vectors. As a
consequence, we prove that unnormalised parameteric inference in non-IID models
can be turned into a semi-parametric estimation problem. Moreover, we show that
the noise-contrastive divergence of Gutmann & Hyv\"arinen (2012) can be
understood as an approximation of the Poisson transform, and extended to
non-IID settings. We use our results to fit spatial Markov chain models of eye
movements, where the Poisson transform allows us to turn a highly non-standard
model into vanilla semi-parametric logistic regression
Entropy rigidity of Hilbert and Riemannian metrics
In this paper we provide two new characterizations of real hyperbolic
-space using the Poincar\'e exponent of a discrete group and the volume
growth entropy. The first characterization is in the space of Hilbert metrics
and generalizes a result of Crampon. The second is in the space of Riemannian
metrics with Ricci curvature bounded below and generalizes a result of
Ledrappier and Wang.Comment: 14 pages, some revisions following the referees remarks. To be
published in IMR
Divide and conquer in ABC: Expectation-Progagation algorithms for likelihood-free inference
ABC algorithms are notoriously expensive in computing time, as they require
simulating many complete artificial datasets from the model. We advocate in
this paper a "divide and conquer" approach to ABC, where we split the
likelihood into n factors, and combine in some way n "local" ABC approximations
of each factor. This has two advantages: (a) such an approach is typically much
faster than standard ABC and (b) it makes it possible to use local summary
statistics (i.e. summary statistics that depend only on the data-points that
correspond to a single factor), rather than global summary statistics (that
depend on the complete dataset). This greatly alleviates the bias introduced by
summary statistics, and even removes it entirely in situations where local
summary statistics are simply the identity function.
We focus on EP (Expectation-Propagation), a convenient and powerful way to
combine n local approximations into a global approximation. Compared to the EP-
ABC approach of Barthelm\'e and Chopin (2014), we present two variations, one
based on the parallel EP algorithm of Cseke and Heskes (2011), which has the
advantage of being implementable on a parallel architecture, and one version
which bridges the gap between standard EP and parallel EP. We illustrate our
approach with an expensive application of ABC, namely inference on spatial
extremes.Comment: To appear in the forthcoming Handbook of Approximate Bayesian
Computation (ABC), edited by S. Sisson, L. Fan, and M. Beaumon
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