116,674 research outputs found
Estimating Local Function Complexity via Mixture of Gaussian Processes
Real world data often exhibit inhomogeneity, e.g., the noise level, the
sampling distribution or the complexity of the target function may change over
the input space. In this paper, we try to isolate local function complexity in
a practical, robust way. This is achieved by first estimating the locally
optimal kernel bandwidth as a functional relationship. Specifically, we propose
Spatially Adaptive Bandwidth Estimation in Regression (SABER), which employs
the mixture of experts consisting of multinomial kernel logistic regression as
a gate and Gaussian process regression models as experts. Using the locally
optimal kernel bandwidths, we deduce an estimate to the local function
complexity by drawing parallels to the theory of locally linear smoothing. We
demonstrate the usefulness of local function complexity for model
interpretation and active learning in quantum chemistry experiments and fluid
dynamics simulations.Comment: 19 pages, 16 figure
Estimating linear functionals in nonlinear regression with responses missing at random
We consider regression models with parametric (linear or nonlinear)
regression function and allow responses to be ``missing at random.'' We assume
that the errors have mean zero and are independent of the covariates. In order
to estimate expectations of functions of covariate and response we use a fully
imputed estimator, namely an empirical estimator based on estimators of
conditional expectations given the covariate. We exploit the independence of
covariates and errors by writing the conditional expectations as unconditional
expectations, which can now be estimated by empirical plug-in estimators. The
mean zero constraint on the error distribution is exploited by adding suitable
residual-based weights. We prove that the estimator is efficient (in the sense
of H\'{a}jek and Le Cam) if an efficient estimator of the parameter is used.
Our results give rise to new efficient estimators of smooth transformations of
expectations. Estimation of the mean response is discussed as a special
(degenerate) case.Comment: Published in at http://dx.doi.org/10.1214/08-AOS642 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Control of Complex Dynamic Systems by Neural Networks
This paper considers the use of neural networks (NN's) in controlling a nonlinear, stochastic system with unknown process equations. The NN is used to model the resulting unknown control law. The approach here is based on using the output error of the system to train the NN controller without the need to construct a separate model (NN or other type) for the unknown process dynamics. To implement such a direct adaptive control approach, it is required that connection weights in the NN be estimated while the system is being controlled. As a result of the feedback of the unknown process dynamics, however, it is not possible to determine the gradient of the loss function for use in standard (back-propagation-type) weight estimation algorithms. Therefore, this paper considers the use of a new stochastic approximation algorithm for this weight estimation, which is based on a 'simultaneous perturbation' gradient approximation that only requires the system output error. It is shown that this algorithm can greatly enhance the efficiency over more standard stochastic approximation algorithms based on finite-difference gradient approximations
A flexible space-variant anisotropic regularisation for image restoration with automated parameter selection
We propose a new space-variant anisotropic regularisation term for
variational image restoration, based on the statistical assumption that the
gradients of the target image distribute locally according to a bivariate
generalised Gaussian distribution. The highly flexible variational structure of
the corresponding regulariser encodes several free parameters which hold the
potential for faithfully modelling the local geometry in the image and
describing local orientation preferences. For an automatic estimation of such
parameters, we design a robust maximum likelihood approach and report results
on its reliability on synthetic data and natural images. For the numerical
solution of the corresponding image restoration model, we use an iterative
algorithm based on the Alternating Direction Method of Multipliers (ADMM). A
suitable preliminary variable splitting together with a novel result in
multivariate non-convex proximal calculus yield a very efficient minimisation
algorithm. Several numerical results showing significant quality-improvement of
the proposed model with respect to some related state-of-the-art competitors
are reported, in particular in terms of texture and detail preservation
Self-similar prior and wavelet bases for hidden incompressible turbulent motion
This work is concerned with the ill-posed inverse problem of estimating
turbulent flows from the observation of an image sequence. From a Bayesian
perspective, a divergence-free isotropic fractional Brownian motion (fBm) is
chosen as a prior model for instantaneous turbulent velocity fields. This
self-similar prior characterizes accurately second-order statistics of velocity
fields in incompressible isotropic turbulence. Nevertheless, the associated
maximum a posteriori involves a fractional Laplacian operator which is delicate
to implement in practice. To deal with this issue, we propose to decompose the
divergent-free fBm on well-chosen wavelet bases. As a first alternative, we
propose to design wavelets as whitening filters. We show that these filters are
fractional Laplacian wavelets composed with the Leray projector. As a second
alternative, we use a divergence-free wavelet basis, which takes implicitly
into account the incompressibility constraint arising from physics. Although
the latter decomposition involves correlated wavelet coefficients, we are able
to handle this dependence in practice. Based on these two wavelet
decompositions, we finally provide effective and efficient algorithms to
approach the maximum a posteriori. An intensive numerical evaluation proves the
relevance of the proposed wavelet-based self-similar priors.Comment: SIAM Journal on Imaging Sciences, 201
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