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3 pages, 173 KiB

Open AccessEditorial

Lattice Computing: A Mathematical Modelling Paradigm for Cyber-Physical System Applications

by Vassilis G. Kaburlasos

Mathematics 2022, 10(2), 271; https://doi.org/10.3390/math10020271 - 16 Jan 2022

Cited by 6 | Viewed by 1637

Abstract

By “model”, we mean a mathematical description of a world aspect [...] Full article

(This article belongs to the Special Issue Lattice Computing: A Mathematical Modelling Paradigm for Cyber–Physical System Applications)

Research

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23 pages, 3819 KiB

Open AccessEditor’s ChoiceArticle

Granule-Based-Classifier (GbC): A Lattice Computing Scheme Applied on Tree Data Structures

by Vassilis G. Kaburlasos, Chris Lytridis, Eleni Vrochidou, Christos Bazinas, George A. Papakostas, Anna Lekova, Omar Bouattane, Mohamed Youssfi and Takashi Hashimoto

Mathematics 2021, 9(22), 2889; https://doi.org/10.3390/math9222889 - 13 Nov 2021

Cited by 8 | Viewed by 1856

Abstract

Social robots keep proliferating. A critical challenge remains their sensible interaction with humans, especially in real world applications. Hence, computing with real world semantics is instrumental. Recently, the Lattice Computing (LC) paradigm has been proposed with a capacity to compute with semantics represented [...] Read more.

Social robots keep proliferating. A critical challenge remains their sensible interaction with humans, especially in real world applications. Hence, computing with real world semantics is instrumental. Recently, the Lattice Computing (LC) paradigm has been proposed with a capacity to compute with semantics represented by partial order in a mathematical lattice data domain. In the aforementioned context, this work proposes a parametric LC classifier, namely a Granule-based-Classifier (GbC), applicable in a mathematical lattice (T,⊑) of tree data structures, each of which represents a human face. A tree data structure here emerges from 68 facial landmarks (points) computed in a data preprocessing step by the OpenFace software. The proposed (tree) representation retains human anonymity during data processing. Extensive computational experiments regarding three different pattern recognition problems, namely (1) head orientation, (2) facial expressions, and (3) human face recognition, demonstrate GbC capacities, including good classification results, and a common human face representation in different pattern recognition problems, as well as data induced granular rules in (T,⊑) that allow for (a) explainable decision-making, (b) tunable generalization enabled also by formal logic/reasoning techniques, and (c) an inherent capacity for modular data fusion extensions. The potential of the proposed techniques is discussed. Full article

(This article belongs to the Special Issue Lattice Computing: A Mathematical Modelling Paradigm for Cyber–Physical System Applications)

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22 pages, 1706 KiB

Open AccessArticle

On Machine-Learning Morphological Image Operators

by Nina S. T. Hirata and George A. Papakostas

Mathematics 2021, 9(16), 1854; https://doi.org/10.3390/math9161854 - 05 Aug 2021

Cited by 7 | Viewed by 2806

Abstract

Morphological operators are nonlinear transformations commonly used in image processing. Their theoretical foundation is based on lattice theory, and it is a well-known result that a large class of image operators can be expressed in terms of two basic ones, the erosions and [...] Read more.

Morphological operators are nonlinear transformations commonly used in image processing. Their theoretical foundation is based on lattice theory, and it is a well-known result that a large class of image operators can be expressed in terms of two basic ones, the erosions and the dilations. In practice, useful operators can be built by combining these two operators, and the new operators can be further combined to implement more complex transformations. The possibility of implementing a compact combination that performs a complex transformation of images is particularly appealing in resource-constrained hardware scenarios. However, finding a proper combination may require a considerable trial-and-error effort. This difficulty has motivated the development of machine-learning-based approaches for designing morphological image operators. In this work, we present an overview of this topic, divided in three parts. First, we review and discuss the representation structure of morphological image operators. Then we address the problem of learning morphological image operators from data, and how representation manifests in the formulation of this problem as well as in the learned operators. In the last part we focus on recent morphological image operator learning methods that take advantage of deep-learning frameworks. We close with discussions and a list of prospective future research directions. Full article

(This article belongs to the Special Issue Lattice Computing: A Mathematical Modelling Paradigm for Cyber–Physical System Applications)

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42 pages, 579 KiB

Open AccessArticle

Four-Fold Formal Concept Analysis Based on Complete Idempotent Semifields

by Francisco José Valverde-Albacete and Carmen Peláez-Moreno

Mathematics 2021, 9(2), 173; https://doi.org/10.3390/math9020173 - 15 Jan 2021

Cited by 4 | Viewed by 1635

Abstract

Formal Concept Analysis (FCA) is a well-known supervised boolean data-mining technique rooted in Lattice and Order Theory, that has several extensions to, e.g., fuzzy and idempotent semirings. At the heart of FCA lies a Galois connection between two powersets. In this paper we [...] Read more.

Formal Concept Analysis (FCA) is a well-known supervised boolean data-mining technique rooted in Lattice and Order Theory, that has several extensions to, e.g., fuzzy and idempotent semirings. At the heart of FCA lies a Galois connection between two powersets. In this paper we extend the FCA formalism to include all four Galois connections between four different semivectors spaces over idempotent semifields, at the same time. The result is

\bar{K}

-four-fold Formal Concept Analysis (

\bar{K}

-4FCA) where

\bar{K}

is the idempotent semifield biasing the analysis. Since complete idempotent semifields come in dually-ordered pairs—e.g., the complete max-plus and min-plus semirings—the basic construction shows dual-order-, row–column- and Galois-connection-induced dualities that appear simultaneously a number of times to provide the full spectrum of variability. Our results lead to a fundamental theorem of

\bar{K}

-four-fold Formal Concept Analysis that properly defines quadrilattices as 4-tuples of (order-dually) isomorphic lattices of vectors and discuss its relevance vis-à-vis previous formal conceptual analyses and some affordances of their results. Full article

(This article belongs to the Special Issue Lattice Computing: A Mathematical Modelling Paradigm for Cyber–Physical System Applications)

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39 pages, 47181 KiB

Open AccessArticle

The Singular Value Decomposition over Completed Idempotent Semifields

by Francisco J. Valverde-Albacete and Carmen Peláez-Moreno

Mathematics 2020, 8(9), 1577; https://doi.org/10.3390/math8091577 - 12 Sep 2020

Cited by 5 | Viewed by 2437

Abstract

In this paper, we provide a basic technique for Lattice Computing: an analogue of the Singular Value Decomposition for rectangular matrices over complete idempotent semifields (i-SVD). These algebras are already complete lattices and many of their instances—the complete schedule algebra or completed max-plus [...] Read more.

In this paper, we provide a basic technique for Lattice Computing: an analogue of the Singular Value Decomposition for rectangular matrices over complete idempotent semifields (i-SVD). These algebras are already complete lattices and many of their instances—the complete schedule algebra or completed max-plus semifield, the tropical algebra, and the max-times algebra—are useful in a range of applications, e.g., morphological processing. We further the task of eliciting the relation between i-SVD and the extension of Formal Concept Analysis to complete idempotent semifields (K-FCA) started in a prior work. We find out that for a matrix with entries considered in a complete idempotent semifield, the Galois connection at the heart of K-FCA provides two basis of left- and right-singular vectors to choose from, for reconstructing the matrix. These are join-dense or meet-dense sets of object or attribute concepts of the concept lattice created by the connection, and they are almost surely not pairwise orthogonal. We conclude with an attempt analogue of the fundamental theorem of linear algebra that gathers all results and discuss it in the wider setting of matrix factorization. Full article

(This article belongs to the Special Issue Lattice Computing: A Mathematical Modelling Paradigm for Cyber–Physical System Applications)

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18 pages, 750 KiB

Open AccessArticle

by Gerhard X. Ritter, Gonzalo Urcid and Luis-David Lara-Rodríguez

Mathematics 2020, 8(9), 1439; https://doi.org/10.3390/math8091439 - 27 Aug 2020

Cited by 1 | Viewed by 1799

Abstract

This paper presents a novel lattice based biomimetic neural network trained by means of a similarity measure derived from a lattice positive valuation. For a wide class of pattern recognition problems, the proposed artificial neural network, implemented as a dendritic hetero-associative memory delivers [...] Read more.

This paper presents a novel lattice based biomimetic neural network trained by means of a similarity measure derived from a lattice positive valuation. For a wide class of pattern recognition problems, the proposed artificial neural network, implemented as a dendritic hetero-associative memory delivers high percentages of successful classification. The memory is a feedforward dendritic network whose arithmetical operations are based on lattice algebra and can be applied to real multivalued inputs. In this approach, the realization of recognition tasks, shows the inherent capability of prototype-class pattern associations in a fast and straightforward manner without need of any iterative scheme subject to issues about convergence. Using an artificially designed data set we show how the proposed trained neural net classifies a test input pattern. Application to a few typical real-world data sets illustrate the overall network classification performance using different training and testing sample subsets generated randomly. Full article

(This article belongs to the Special Issue Lattice Computing: A Mathematical Modelling Paradigm for Cyber–Physical System Applications)

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21 pages, 2875 KiB

Open AccessArticle

Reduced Dilation-Erosion Perceptron for Binary Classification

by Marcos Eduardo Valle

Mathematics 2020, 8(4), 512; https://doi.org/10.3390/math8040512 - 02 Apr 2020

Cited by 12 | Viewed by 2316

Abstract

Dilation and erosion are two elementary operations from mathematical morphology, a non-linear lattice computing methodology widely used for image processing and analysis. The dilation-erosion perceptron (DEP) is a morphological neural network obtained by a convex combination of a dilation and an erosion followed [...] Read more.

Dilation and erosion are two elementary operations from mathematical morphology, a non-linear lattice computing methodology widely used for image processing and analysis. The dilation-erosion perceptron (DEP) is a morphological neural network obtained by a convex combination of a dilation and an erosion followed by the application of a hard-limiter function for binary classification tasks. A DEP classifier can be trained using a convex-concave procedure along with the minimization of the hinge loss function. As a lattice computing model, the DEP classifier assumes the feature and class spaces are partially ordered sets. In many practical situations, however, there is no natural ordering for the feature patterns. Using concepts from multi-valued mathematical morphology, this paper introduces the reduced dilation-erosion (r-DEP) classifier. An r-DEP classifier is obtained by endowing the feature space with an appropriate reduced ordering. Such reduced ordering can be determined using two approaches: one based on an ensemble of support vector classifiers (SVCs) with different kernels and the other based on a bagging of similar SVCs trained using different samples of the training set. Using several binary classification datasets from the OpenML repository, the ensemble and bagging r-DEP classifiers yielded mean higher balanced accuracy scores than the linear, polynomial, and radial basis function (RBF) SVCs as well as their ensemble and a bagging of RBF SVCs. Full article

(This article belongs to the Special Issue Lattice Computing: A Mathematical Modelling Paradigm for Cyber–Physical System Applications)

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14 pages, 2111 KiB

Open AccessArticle

WINkNN: Windowed Intervals’ Number kNN Classifier for Efficient Time-Series Applications

by Chris Lytridis, Anna Lekova, Christos Bazinas, Michail Manios and Vassilis G. Kaburlasos

Mathematics 2020, 8(3), 413; https://doi.org/10.3390/math8030413 - 13 Mar 2020

Cited by 14 | Viewed by 2291

Abstract

Our interest is in time series classification regarding cyber–physical systems (CPSs) with emphasis in human-robot interaction. We propose an extension of the k nearest neighbor (kNN) classifier to time-series classification using intervals’ numbers (INs). More specifically, we partition a time-series into windows of [...] Read more.

Our interest is in time series classification regarding cyber–physical systems (CPSs) with emphasis in human-robot interaction. We propose an extension of the k nearest neighbor (kNN) classifier to time-series classification using intervals’ numbers (INs). More specifically, we partition a time-series into windows of equal length and from each window data we induce a distribution which is represented by an IN. This preserves the time dimension in the representation. All-order data statistics, represented by an IN, are employed implicitly as features; moreover, parametric non-linearities are introduced in order to tune the geometrical relationship (i.e., the distance) between signals and consequently tune classification performance. In conclusion, we introduce the windowed IN kNN (WINkNN) classifier whose application is demonstrated comparatively in two benchmark datasets regarding, first, electroencephalography (EEG) signals and, second, audio signals. The results by WINkNN are superior in both problems; in addition, no ad-hoc data preprocessing is required. Potential future work is discussed. Full article

(This article belongs to the Special Issue Lattice Computing: A Mathematical Modelling Paradigm for Cyber–Physical System Applications)

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