2,660 research outputs found
A proof of convergence of multi-class logistic regression network
This paper revisits the special type of a neural network known under two
names. In the statistics and machine learning community it is known as a
multi-class logistic regression neural network. In the neural network
community, it is simply the soft-max layer. The importance is underscored by
its role in deep learning: as the last layer, whose autput is actually the
classification of the input patterns, such as images. Our exposition focuses on
mathematically rigorous derivation of the key equation expressing the gradient.
The fringe benefit of our approach is a fully vectorized expression, which is a
basis of an efficient implementation. The second result of this paper is the
positivity of the second derivative of the cross-entropy loss function as
function of the weights. This result proves that optimization methods based on
convexity may be used to train this network. As a corollary, we demonstrate
that no -regularizer is needed to guarantee convergence of gradient
descent
Deformed SPDE models with an application to spatial modeling of significant wave height
A non-stationary Gaussian random field model is developed based on a
combination of the stochastic partial differential equation (SPDE) approach and
the classical deformation method. With the deformation method, a stationary
field is defined on a domain which is deformed so that the field becomes
non-stationary. We show that if the stationary field is a Mat'ern field defined
as a solution to a fractional SPDE, the resulting non-stationary model can be
represented as the solution to another fractional SPDE on the deformed domain.
By defining the model in this way, the computational advantages of the SPDE
approach can be combined with the deformation method's more intuitive
parameterisation of non-stationarity. In particular it allows for independent
control over the non-stationary practical correlation range and the variance,
which has not been possible with previously proposed non-stationary SPDE
models.
The model is tested on spatial data of significant wave height, a
characteristic of ocean surface conditions which is important when estimating
the wear and risks associated with a planned journey of a ship. The model
parameters are estimated to data from the north Atlantic using a maximum
likelihood approach. The fitted model is used to compute wave height exceedance
probabilities and the distribution of accumulated fatigue damage for ships
traveling a popular shipping route. The model results agree well with the data,
indicating that the model could be used for route optimization in naval
logistics.Comment: 22 pages, 12 figure
Error reduction in density estimation under shape restrictions
For the problems of nonparametric estimation of nonincreasing and symmetric unimodal density functions with bounded supports we determine the projections of estimates onto the convex families of possible parent densities with respect to the weighted integrated squared error. We also describe the method of approximating the analogous projections onto the respective density classes satisfying some general moment conditions. The method of projections reduces the estimation errors for all possible values of observations of a given finite sample size in a uniformly optimal way and provides estimates sharing the properties of the parent densities
Realizacja praw mniejszości białoruskiej w Polsce i mniejszości polskiej na Białorusi
Auror opisuje sytuację mniejszości białoruskiej w Polsce i mniejszości polskiej na Białorusi, w zakresie: liczebności i rozmieszczenia, organizacji zrzeszających mniejszości, działalności kulturalnej, występowania w mediach, oświaty, życia politycznego, religijnego, języka i uregulowań prawnych.Udostępnienie publikacji Wydawnictwa Uniwersytetu Łódzkiego finansowane w ramach projektu „Doskonałość naukowa kluczem do doskonałości kształcenia”. Projekt realizowany jest ze środków Europejskiego Funduszu Społecznego w ramach Programu Operacyjnego Wiedza Edukacja Rozwój; nr umowy: POWER.03.05.00-00-Z092/17-00
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