10,379 research outputs found
Orthonormal bases of regular wavelets in spaces of homogeneous type
Adapting the recently developed randomized dyadic structures, we introduce
the notion of spline function in geometrically doubling quasi-metric spaces.
Such functions have interpolation and reproducing properties as the linear
splines in Euclidean spaces. They also have H\"older regularity. This is used
to build an orthonormal basis of H\"older-continuous wavelets with exponential
decay in any space of homogeneous type. As in the classical theory, wavelet
bases provide a universal Calder\'on reproducing formula to study and develop
function space theory and singular integrals. We discuss the examples of
spaces, BMO and apply this to a proof of the T(1) theorem. As no extra
condition {(like 'reverse doubling', 'small boundary' of balls, etc.)} on the
space of homogeneous type is required, our results extend a long line of works
on the subject.Comment: We have made improvements to section 2 following the referees
suggestions. In particular, it now contains full proof of formerly Theorem
2.7 instead of sending back to earlier works, which makes the construction of
splines self-contained. One reference adde
Optimising Spatial and Tonal Data for PDE-based Inpainting
Some recent methods for lossy signal and image compression store only a few
selected pixels and fill in the missing structures by inpainting with a partial
differential equation (PDE). Suitable operators include the Laplacian, the
biharmonic operator, and edge-enhancing anisotropic diffusion (EED). The
quality of such approaches depends substantially on the selection of the data
that is kept. Optimising this data in the domain and codomain gives rise to
challenging mathematical problems that shall be addressed in our work.
In the 1D case, we prove results that provide insights into the difficulty of
this problem, and we give evidence that a splitting into spatial and tonal
(i.e. function value) optimisation does hardly deteriorate the results. In the
2D setting, we present generic algorithms that achieve a high reconstruction
quality even if the specified data is very sparse. To optimise the spatial
data, we use a probabilistic sparsification, followed by a nonlocal pixel
exchange that avoids getting trapped in bad local optima. After this spatial
optimisation we perform a tonal optimisation that modifies the function values
in order to reduce the global reconstruction error. For homogeneous diffusion
inpainting, this comes down to a least squares problem for which we prove that
it has a unique solution. We demonstrate that it can be found efficiently with
a gradient descent approach that is accelerated with fast explicit diffusion
(FED) cycles. Our framework allows to specify the desired density of the
inpainting mask a priori. Moreover, is more generic than other data
optimisation approaches for the sparse inpainting problem, since it can also be
extended to nonlinear inpainting operators such as EED. This is exploited to
achieve reconstructions with state-of-the-art quality.
We also give an extensive literature survey on PDE-based image compression
methods
Urn Models and Beta-splines
Some insight into the properties of beta-splines is gained by applying the techniques of urn models. Urn models are used to construct beta-spline basis functions and to derive the basic properties of these blending functions and the corresponding beta-spline curves. Only the simple notion of linear geometric continuity and with the most elementary beta parameter are outlined. Non-linear geometric continuity leads to additional beta parameters and to more complicated basis functions. Whether urn models can give us any insight into these higher order concepts still remains to be investigated
Probabilistic Line Searches for Stochastic Optimization
In deterministic optimization, line searches are a standard tool ensuring
stability and efficiency. Where only stochastic gradients are available, no
direct equivalent has so far been formulated, because uncertain gradients do
not allow for a strict sequence of decisions collapsing the search space. We
construct a probabilistic line search by combining the structure of existing
deterministic methods with notions from Bayesian optimization. Our method
retains a Gaussian process surrogate of the univariate optimization objective,
and uses a probabilistic belief over the Wolfe conditions to monitor the
descent. The algorithm has very low computational cost, and no user-controlled
parameters. Experiments show that it effectively removes the need to define a
learning rate for stochastic gradient descent.Comment: Extended version of the NIPS '15 conference paper, includes detailed
pseudo-code, 59 pages, 35 figure
Robust EM algorithm for model-based curve clustering
Model-based clustering approaches concern the paradigm of exploratory data
analysis relying on the finite mixture model to automatically find a latent
structure governing observed data. They are one of the most popular and
successful approaches in cluster analysis. The mixture density estimation is
generally performed by maximizing the observed-data log-likelihood by using the
expectation-maximization (EM) algorithm. However, it is well-known that the EM
algorithm initialization is crucial. In addition, the standard EM algorithm
requires the number of clusters to be known a priori. Some solutions have been
provided in [31, 12] for model-based clustering with Gaussian mixture models
for multivariate data. In this paper we focus on model-based curve clustering
approaches, when the data are curves rather than vectorial data, based on
regression mixtures. We propose a new robust EM algorithm for clustering
curves. We extend the model-based clustering approach presented in [31] for
Gaussian mixture models, to the case of curve clustering by regression
mixtures, including polynomial regression mixtures as well as spline or
B-spline regressions mixtures. Our approach both handles the problem of
initialization and the one of choosing the optimal number of clusters as the EM
learning proceeds, rather than in a two-fold scheme. This is achieved by
optimizing a penalized log-likelihood criterion. A simulation study confirms
the potential benefit of the proposed algorithm in terms of robustness
regarding initialization and funding the actual number of clusters.Comment: In Proceedings of the 2013 International Joint Conference on Neural
Networks (IJCNN), 2013, Dallas, TX, US
Forecasting of commercial sales with large scale Gaussian Processes
This paper argues that there has not been enough discussion in the field of
applications of Gaussian Process for the fast moving consumer goods industry.
Yet, this technique can be important as it e.g., can provide automatic feature
relevance determination and the posterior mean can unlock insights on the data.
Significant challenges are the large size and high dimensionality of commercial
data at a point of sale. The study reviews approaches in the Gaussian Processes
modeling for large data sets, evaluates their performance on commercial sales
and shows value of this type of models as a decision-making tool for
management.Comment: 1o pages, 5 figure
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