Showing papers by "Irina Perfilieva published in 2020"

Deep Learning and Higher Degree F-Transforms: Interpretable Kernels Before and After Learning

Abstract: One of the current trends in the deep neural network technology consists in allowing a man–machine interaction and providing an explanation of network design and learning principles. In this direction, an experience with fuzzy systems is of great support. We propose our insight that is based on the particular theory of fuzzy (F)-transforms. Besides a theoretical explanation, we develop a new architecture of a deep neural network where the F-transform convolution kernels are used in the first two layers. Based on a series of experiments, we demonstrate the suitability of the F-transform-based deep neural network in the domain of image processing with the focus on recognition. Moreover, we support our insight by revealing the similarity between the F-transform and first-layer kernels in the most used deep neural networks.

Boundary value problem: Weak solutions induced by fuzzy partitions

Abstract: The aim of this paper is to propose a new methodology in the construction of spaces of test functions used in a weak formulation of the Boundary Value Problem. The proposed construction is based on the so called "two dimensional" approach where at first, we select a partition of a domain and second, a dimension of an approximating functional subspace on each partition element. The main advantage consists in the independent selection of the key parameters, aiming at achieving a requested quality of approximation with a reasonable complexity. We give theoretical justification and illustration on examples that confirm our methodology.

Nonlocal Laplace Operator in a Space with the Fuzzy Partition

Abstract: Differential operators play an important role in the mathematical modeling of dynamic processes and the analysis of various structures. However, there are certain limitations in their use. To remove them, nonlocal differential operators have been proposed. In this work, we focus on nonlocal Laplace operator, which has become increasingly useful in image processing. We introduce the representation of F-transform based Laplace operator in a space with a fuzzy partition. Many useful properties of this operator are proposed and their proofs are also included.

Measure of Lattice-Valued Direct F-transforms and Its Topological Interpretations

Abstract: The goal is to introduce and study the measure of quality of approximation of a given fuzzy set by its lattice-valued F-transform. Further, we show that this measure is connected with an Alexandroff LM-fuzzy topological (co-topological) spaces. Finally, we discuss the categorical relationship between the defined structures.

Categories of L-Fuzzy Čech Closure Spaces and L-Fuzzy Co-Topological Spaces

Abstract: Recently, fuzzy systems have become one of the hottest topics due to their applications in the area of computer science. Therefore, in this article, we are making efforts to add new useful relationships between the selected L-fuzzy (fuzzifying) systems. In particular, we establish relationships between L-fuzzy (fuzzifying) Cech closure spaces, L-fuzzy (fuzzifying) co-topological spaces and L-fuzzy (fuzzifying) approximation spaces based on reflexive L-fuzzy relations. We also show that there is a Galois correspondence between the categories of these spaces.

F-Transform and Convolutional NN: Cross-Fertilization and Step Forward

Abstract: We propose to assign the F-transform kernels to the CNN weights and compare them with commonly used initialization. By this, we develop a new initialization mechanism where the F-transform convolution kernels are used in the convolutional layers. Based on a series of experiments, we demonstrate the suitability of the F-transform-based deep neural network in the domain of image processing with the focus on classification. Moreover, we support our insight by revealing the similarity between the F-transform and first-layer kernels in certain deep neural networks.

M-Fuzzifying Approximation Spaces, M-Fuzzifying Pretopological Spaces and M-Fuzzifying Closure Spaces

Abstract: In category theory, Galois connection plays a significant role in developing the connections among different structures. The objective of this work is to investigate the essential connections among...

On Categories of L-Fuzzifying Approximation Spaces, L-Fuzzifying Pretopological Spaces and L-Fuzzifying Closure Spaces

Abstract: This paper investigates the essential connections among several categories with a weaker structure than that of L-fuzzifying topology, namely category of L-fuzzifying approximation spaces based on reflexive L-fuzzy relations, category of L-fuzzifying pretopological spaces and category of L-fuzzifying interior (closure) spaces. The interrelations among these structures are established in categorical setup.