1,440 research outputs found
Nonparametric likelihood based estimation of linear filters for point processes
We consider models for multivariate point processes where the intensity is
given nonparametrically in terms of functions in a reproducing kernel Hilbert
space. The likelihood function involves a time integral and is consequently not
given in terms of a finite number of kernel evaluations. The main result is a
representation of the gradient of the log-likelihood, which we use to derive
computable approximations of the log-likelihood and the gradient by time
discretization. These approximations are then used to minimize the approximate
penalized log-likelihood. For time and memory efficiency the implementation
relies crucially on the use of sparse matrices. As an illustration we consider
neuron network modeling, and we use this example to investigate how the
computational costs of the approximations depend on the resolution of the time
discretization. The implementation is available in the R package ppstat.Comment: 10 pages, 3 figure
Neural Class-Specific Regression for face verification
Face verification is a problem approached in the literature mainly using
nonlinear class-specific subspace learning techniques. While it has been shown
that kernel-based Class-Specific Discriminant Analysis is able to provide
excellent performance in small- and medium-scale face verification problems,
its application in today's large-scale problems is difficult due to its
training space and computational requirements. In this paper, generalizing our
previous work on kernel-based class-specific discriminant analysis, we show
that class-specific subspace learning can be cast as a regression problem. This
allows us to derive linear, (reduced) kernel and neural network-based
class-specific discriminant analysis methods using efficient batch and/or
iterative training schemes, suited for large-scale learning problems. We test
the performance of these methods in two datasets describing medium- and
large-scale face verification problems.Comment: 9 pages, 4 figure
A Geometric Variational Approach to Bayesian Inference
We propose a novel Riemannian geometric framework for variational inference
in Bayesian models based on the nonparametric Fisher-Rao metric on the manifold
of probability density functions. Under the square-root density representation,
the manifold can be identified with the positive orthant of the unit
hypersphere in L2, and the Fisher-Rao metric reduces to the standard L2 metric.
Exploiting such a Riemannian structure, we formulate the task of approximating
the posterior distribution as a variational problem on the hypersphere based on
the alpha-divergence. This provides a tighter lower bound on the marginal
distribution when compared to, and a corresponding upper bound unavailable
with, approaches based on the Kullback-Leibler divergence. We propose a novel
gradient-based algorithm for the variational problem based on Frechet
derivative operators motivated by the geometry of the Hilbert sphere, and
examine its properties. Through simulations and real-data applications, we
demonstrate the utility of the proposed geometric framework and algorithm on
several Bayesian models
The discriminative functional mixture model for a comparative analysis of bike sharing systems
Bike sharing systems (BSSs) have become a means of sustainable intermodal
transport and are now proposed in many cities worldwide. Most BSSs also provide
open access to their data, particularly to real-time status reports on their
bike stations. The analysis of the mass of data generated by such systems is of
particular interest to BSS providers to update system structures and policies.
This work was motivated by interest in analyzing and comparing several European
BSSs to identify common operating patterns in BSSs and to propose practical
solutions to avoid potential issues. Our approach relies on the identification
of common patterns between and within systems. To this end, a model-based
clustering method, called FunFEM, for time series (or more generally functional
data) is developed. It is based on a functional mixture model that allows the
clustering of the data in a discriminative functional subspace. This model
presents the advantage in this context to be parsimonious and to allow the
visualization of the clustered systems. Numerical experiments confirm the good
behavior of FunFEM, particularly compared to state-of-the-art methods. The
application of FunFEM to BSS data from JCDecaux and the Transport for London
Initiative allows us to identify 10 general patterns, including pathological
ones, and to propose practical improvement strategies based on the system
comparison. The visualization of the clustered data within the discriminative
subspace turns out to be particularly informative regarding the system
efficiency. The proposed methodology is implemented in a package for the R
software, named funFEM, which is available on the CRAN. The package also
provides a subset of the data analyzed in this work.Comment: Published at http://dx.doi.org/10.1214/15-AOAS861 in the Annals of
Applied Statistics (http://www.imstat.org/aoas/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Positive Definite Kernels in Machine Learning
This survey is an introduction to positive definite kernels and the set of
methods they have inspired in the machine learning literature, namely kernel
methods. We first discuss some properties of positive definite kernels as well
as reproducing kernel Hibert spaces, the natural extension of the set of
functions associated with a kernel defined
on a space . We discuss at length the construction of kernel
functions that take advantage of well-known statistical models. We provide an
overview of numerous data-analysis methods which take advantage of reproducing
kernel Hilbert spaces and discuss the idea of combining several kernels to
improve the performance on certain tasks. We also provide a short cookbook of
different kernels which are particularly useful for certain data-types such as
images, graphs or speech segments.Comment: draft. corrected a typo in figure
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