1,784 research outputs found
A trivariate interpolation algorithm using a cube-partition searching procedure
In this paper we propose a fast algorithm for trivariate interpolation, which
is based on the partition of unity method for constructing a global interpolant
by blending local radial basis function interpolants and using locally
supported weight functions. The partition of unity algorithm is efficiently
implemented and optimized by connecting the method with an effective
cube-partition searching procedure. More precisely, we construct a cube
structure, which partitions the domain and strictly depends on the size of its
subdomains, so that the new searching procedure and, accordingly, the resulting
algorithm enable us to efficiently deal with a large number of nodes.
Complexity analysis and numerical experiments show high efficiency and accuracy
of the proposed interpolation algorithm
Shape Constrained Regularisation by Statistical Multiresolution for Inverse Problems: Asymptotic Analysis
This paper is concerned with a novel regularisation technique for solving
linear ill-posed operator equations in Hilbert spaces from data that is
corrupted by white noise. We combine convex penalty functionals with
extreme-value statistics of projections of the residuals on a given set of
sub-spaces in the image-space of the operator. We prove general consistency and
convergence rate results in the framework of Bregman-divergences which allows
for a vast range of penalty functionals. Various examples that indicate the
applicability of our approach will be discussed. We will illustrate in the
context of signal and image processing that the presented method constitutes a
locally adaptive reconstruction method
Some Theorems for Feed Forward Neural Networks
In this paper we introduce a new method which employs the concept of
"Orientation Vectors" to train a feed forward neural network and suitable for
problems where large dimensions are involved and the clusters are
characteristically sparse. The new method is not NP hard as the problem size
increases. We `derive' the method by starting from Kolmogrov's method and then
relax some of the stringent conditions. We show for most classification
problems three layers are sufficient and the network size depends on the number
of clusters. We prove as the number of clusters increase from N to N+dN the
number of processing elements in the first layer only increases by d(logN), and
are proportional to the number of classes, and the method is not NP hard.
Many examples are solved to demonstrate that the method of Orientation
Vectors requires much less computational effort than Radial Basis Function
methods and other techniques wherein distance computations are required, in
fact the present method increases logarithmically with problem size compared to
the Radial Basis Function method and the other methods which depend on distance
computations e.g statistical methods where probabilistic distances are
calculated. A practical method of applying the concept of Occum's razor to
choose between two architectures which solve the same classification problem
has been described. The ramifications of the above findings on the field of
Deep Learning have also been briefly investigated and we have found that it
directly leads to the existence of certain types of NN architectures which can
be used as a "mapping engine", which has the property of "invertibility", thus
improving the prospect of their deployment for solving problems involving Deep
Learning and hierarchical classification. The latter possibility has a lot of
future scope in the areas of machine learning and cloud computing.Comment: 15 pages 13 figure
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