362 research outputs found
A Broad Class of Discrete-Time Hypercomplex-Valued Hopfield Neural Networks
In this paper, we address the stability of a broad class of discrete-time
hypercomplex-valued Hopfield-type neural networks. To ensure the neural
networks belonging to this class always settle down at a stationary state, we
introduce novel hypercomplex number systems referred to as real-part
associative hypercomplex number systems. Real-part associative hypercomplex
number systems generalize the well-known Cayley-Dickson algebras and real
Clifford algebras and include the systems of real numbers, complex numbers,
dual numbers, hyperbolic numbers, quaternions, tessarines, and octonions as
particular instances. Apart from the novel hypercomplex number systems, we
introduce a family of hypercomplex-valued activation functions called
-projection functions. Broadly speaking, a
-projection function projects the activation potential onto the
set of all possible states of a hypercomplex-valued neuron. Using the theory
presented in this paper, we confirm the stability analysis of several
discrete-time hypercomplex-valued Hopfield-type neural networks from the
literature. Moreover, we introduce and provide the stability analysis of a
general class of Hopfield-type neural networks on Cayley-Dickson algebras
How active perception and attractor dynamics shape perceptual categorization: A computational model
We propose a computational model of perceptual categorization that fuses elements of grounded and sensorimotor theories of cognition with dynamic models of decision-making. We assume that category information consists in anticipated patterns of agent–environment interactions that can be elicited through overt or covert (simulated) eye movements, object manipulation, etc. This information is firstly encoded when category information is acquired, and then re-enacted during perceptual categorization. The perceptual categorization consists in a dynamic competition between attractors that encode the sensorimotor patterns typical of each category; action prediction success counts as ‘‘evidence’’ for a given category and contributes to falling into the corresponding attractor. The evidence accumulation process is guided by an active perception loop, and the active exploration of objects (e.g., visual exploration) aims at eliciting expected sensorimotor patterns that count as evidence for the object category. We present a computational model incorporating these elements and describing action prediction, active perception, and attractor dynamics as key elements of perceptual categorizations. We test the model in three simulated perceptual categorization tasks, and we discuss its relevance for grounded and sensorimotor theories of cognition.Peer reviewe
Global stability of Clifford-valued Takagi-Sugeno fuzzy neural networks with time-varying delays and impulses
summary:In this study, we consider the Takagi-Sugeno (T-S) fuzzy model to examine the global asymptotic stability of Clifford-valued neural networks with time-varying delays and impulses. In order to achieve the global asymptotic stability criteria, we design a general network model that includes quaternion-, complex-, and real-valued networks as special cases. First, we decompose the -dimensional Clifford-valued neural network into -dimensional real-valued counterparts in order to solve the noncommutativity of Clifford numbers multiplication. Then, we prove the new global asymptotic stability criteria by constructing an appropriate Lyapunov-Krasovskii functionals (LKFs) and employing Jensen's integral inequality together with the reciprocal convex combination method. All the results are proven using linear matrix inequalities (LMIs). Finally, a numerical example is provided to show the effectiveness of the achieved results
Selected aspects of complex, hypercomplex and fuzzy neural networks
This short report reviews the current state of the research and methodology
on theoretical and practical aspects of Artificial Neural Networks (ANN). It
was prepared to gather state-of-the-art knowledge needed to construct complex,
hypercomplex and fuzzy neural networks.
The report reflects the individual interests of the authors and, by now
means, cannot be treated as a comprehensive review of the ANN discipline.
Considering the fast development of this field, it is currently impossible to
do a detailed review of a considerable number of pages.
The report is an outcome of the Project 'The Strategic Research Partnership
for the mathematical aspects of complex, hypercomplex and fuzzy neural
networks' meeting at the University of Warmia and Mazury in Olsztyn, Poland,
organized in September 2022.Comment: 46 page
Extending the Universal Approximation Theorem for a Broad Class of Hypercomplex-Valued Neural Networks
The universal approximation theorem asserts that a single hidden layer neural
network approximates continuous functions with any desired precision on compact
sets. As an existential result, the universal approximation theorem supports
the use of neural networks for various applications, including regression and
classification tasks. The universal approximation theorem is not limited to
real-valued neural networks but also holds for complex, quaternion, tessarines,
and Clifford-valued neural networks. This paper extends the universal
approximation theorem for a broad class of hypercomplex-valued neural networks.
Precisely, we first introduce the concept of non-degenerate hypercomplex
algebra. Complex numbers, quaternions, and tessarines are examples of
non-degenerate hypercomplex algebras. Then, we state the universal
approximation theorem for hypercomplex-valued neural networks defined on a
non-degenerate algebra
- …