On Quaternionic Measures

Luna-Elizarrarás, M. Elena; Pogorui, A. A.; Shapiro, Michael; Kolomiiets, Tamila

doi:10.1007/s00006-020-01090-8

Cited by 3 publications

(1 citation statement)

References 12 publications

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“…Future directions: Future works should introduce novel fully quaternion pooling methods, emphasizing the use of interchannel relationships. For example: using polar representations, introducing quaternion measures [75], or taking inspiration from information theory.…”

Section: Quaternion Poolingmentioning

confidence: 99%

Quaternion Quantum Neural Network for Classification

Altamirano-Escobedo

Bayro–Corrochano

2023

Adv. Appl. Clifford Algebras

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Since their first applications, Convolutional Neural Networks (CNNs) have solved problems that have advanced the state-of-the-art in several domains. CNNs represent information using real numbers. Despite encouraging results, theoretical analysis shows that representations such as hyper-complex numbers can achieve richer representational capacities than real numbers, and that Hamilton products can capture intrinsic interchannel relationships. Moreover, in the last few years, experimental research has shown that Quaternion-Valued CNNs (QCNNs) can achieve similar performance with fewer parameters than their real-valued counterparts. This paper condenses research in the development of QCNNs from its very beginnings. We propose a conceptual organization of current trends and analyze the main building blocks used in the design of QCNN models. Based on this conceptual organization, we propose future directions of research.

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Section: Quaternion Poolingmentioning

confidence: 99%

Quaternion Quantum Neural Network for Classification

Altamirano-Escobedo

Bayro–Corrochano

2023

Adv. Appl. Clifford Algebras

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Quaternion Convolutional Neural Networks: Current Advances and Future Directions

Altamirano-Gomez,

Gershenson

2024

Adv. Appl. Clifford Algebras

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Since their first applications, Convolutional Neural Networks (CNNs) have solved problems that have advanced the state-of-the-art in several domains. CNNs represent information using real numbers. Despite encouraging results, theoretical analysis shows that representations such as hyper-complex numbers can achieve richer representational capacities than real numbers, and that Hamilton products can capture intrinsic interchannel relationships. Moreover, in the last few years, experimental research has shown that Quaternion-valued CNNs (QCNNs) can achieve similar performance with fewer parameters than their real-valued counterparts. This paper condenses research in the development of QCNNs from its very beginnings. We propose a conceptual organization of current trends and analyze the main building blocks used in the design of QCNN models. Based on this conceptual organization, we propose future directions of research.

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Elements of probability theory with values in bihyperbolic algebra

Kolomiiets

2021

Proc. IAMM NASU

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In this paper we expand the concept of a really significant probabilistic measure in the case when the measure takes values in the algebra of bihyperbolic numbers. The basic properties of bihyperbolic numbers are given, in particular idempotents, main ideals generated by idempotents, Pierce's decompo\-sition and the set of zero divisors of the algebra of bihyperbolic numbers are determined. We entered the relation of partial order on the set of bihyperbolic numbers, by means of which the bihyperbolic significant modulus is defined and its basic properties are proved. In addition, some bihyperbolic modules can be endowed with a bihyperbolic significant norms that take values in a set of non-negative bihyperbolic numbers. We define $\sigma$-additive functions of sets in a measurable space that take appropriately normalized bihyperbolic values, which we call a bihyperbolic significant probability. It is proved that such a bihyperbolic probability satisfies the basic properties of the classical probability. A representation of the bihyperbolic probability measure is given and its main properties are proved. A bihyperbolically significant random variable is defined on a bihyperbolic probability space, and this variable is a bihyperbolic measurable function in the same space. We proved the criterion of measurability of a function with values in the algebra of bihyperbolic numbers, and the basic properties of bihyperbolic random variables are formulated and proved. Special cases have been studied in which the bihyperbolic probability and the bihyperbolic random variable take values that are zero divisors of bihyperbolic algebra. Although bihyperbolic numbers are less popular than hyperbolic numbers, bicomplex numbers, or quaternions, they have a number of important properties that can be useful, particularly in the study of partial differential equations also in mathematical statistics for testing complex hypotheses, in thermodynamics and statistical physics.

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On Quaternionic Measures

Cited by 3 publications

References 12 publications

Quaternion Quantum Neural Network for Classification

Quaternion Quantum Neural Network for Classification

Quaternion Convolutional Neural Networks: Current Advances and Future Directions

Elements of probability theory with values in bihyperbolic algebra

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