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Optimal estimators in learning theory

V. N. Temlyakov (2006)

Banach Center Publications

This paper is a survey of recent results on some problems of supervised learning in the setting formulated by Cucker and Smale. Supervised learning, or learning-from-examples, refers to a process that builds on the base of available data of inputs x i and outputs y i , i = 1,...,m, a function that best represents the relation between the inputs x ∈ X and the corresponding outputs y ∈ Y. The goal is to find an estimator f z on the base of given data z : = ( ( x ₁ , y ₁ ) , . . . , ( x m , y m ) ) that approximates well the regression function f ρ of...

Optimization of the maximum likelihood estimator for determining the intrinsic dimensionality of high-dimensional data

Rasa Karbauskaitė, Gintautas Dzemyda (2015)

International Journal of Applied Mathematics and Computer Science

One of the problems in the analysis of the set of images of a moving object is to evaluate the degree of freedom of motion and the angle of rotation. Here the intrinsic dimensionality of multidimensional data, characterizing the set of images, can be used. Usually, the image may be represented by a high-dimensional point whose dimensionality depends on the number of pixels in the image. The knowledge of the intrinsic dimensionality of a data set is very useful information in exploratory data analysis,...

OWA operators for discrete gradual intervals: implications to fuzzy intervals and multi-expert decision making

Zdenko Takáč (2016)

Kybernetika

A new concept in fuzzy sets theory, namely that of gradual element, was introduced recently. It is known that the set of gradual real numbers is not ordered linearly. We restrict our attention to a discrete case and propose a class of linear orders for discrete gradual real numbers. Then, using idea of the so-called admissible order of intervals, we present a class of linear orders for discrete gradual intervals. Once we have the linear orders it is possible to define OWA operator for discrete gradual...

Parametric families of fuzzy consequence operators.

Javier Elorza, Pedro Burillo (2004)

Mathware and Soft Computing

In a previous paper we explored the notion of coherent fuzzy consequence operator. Since we did not know of any example in the literature of non-coherent fuzzy consequence operator, we also showed several families of such operators. It is well-known that the operator induced by a fuzzy preorder through Zadeh's compositional rule is always a coherent fuzzy consequence operator. It is also known that the relation induced by a fuzzy consequence operator is a fuzzy preorder if such operator is coherent....

Partitioned iterated function systems with division and a fractal dependence graph in recognition of 2D shapes

Krzysztof Gdawiec, Diana Domańska (2011)

International Journal of Applied Mathematics and Computer Science

One of the approaches in pattern recognition is the use of fractal geometry. The property of self-similarity of fractals has been used as a feature in several pattern recognition methods. All fractal recognition methods use global analysis of the shape. In this paper we present some drawbacks of these methods and propose fractal local analysis using partitioned iterated function systems with division. Moreover, we introduce a new fractal recognition method based on a dependence graph obtained from...

Phenotypic evolution with a mutation based on symmetric α-stable distributions

Andrzej Obuchowicz, Przemysław Prętki (2004)

International Journal of Applied Mathematics and Computer Science

Multidimensional Symmetric α-Stable (SαS) mutations are applied to phenotypic evolutionary algorithms. Such mutations are characterized by non-spherical symmetry for α<2 and the fact that the most probable distance of mutated points is not in a close neighborhood of the origin, but at a certain distance from it. It is the so-called surrounding effect (Obuchowicz, 2001b; 2003b). For α=2, the SαS mutation reduces to the Gaussian one, and in the case of α=1, the Cauchy mutation is obtained. The...

Picture languages in automatic radiological palm interpretation

Ryszard Tadeusiewicz, Marek Ogiela (2005)

International Journal of Applied Mathematics and Computer Science

The paper presents a new technique for cognitive analysis and recognition of pathological wrist bone lesions. This method uses AI techniques and mathematical linguistics allowing us to automatically evaluate the structure of the said bones, based on palm radiological images. Possibilities of computer interpretation of selected images, based on the methodology of automatic medical image understanding, as introduced by the authors, were created owing to the introduction of an original relational description...

Piecewise approximation and neural networks

Martina Révayová, Csaba Török (2007)

Kybernetika

The paper deals with the recently proposed autotracking piecewise cubic approximation (APCA) based on the discrete projective transformation, and neural networks (NN). The suggested new approach facilitates the analysis of data with complex dependence and relatively small errors. We introduce a new representation of polynomials that can provide different local approximation models. We demonstrate how APCA can be applied to especially noisy data thanks to NN and local estimations. On the other hand,...

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