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 B-projection functions. Broadly speaking, a B-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.

Max-plus and min-plus projection autoassociative morphological memories and their compositions for pattern classification.

Autoassociative morphological memories (AMMs) are robust and computationally efficient memory models with unlimited storage capacity. In this paper, we present the max-plus and min-plus projection autoassociative morphological memories (PAMMs) as well as their compositions. Briefly, the max-plus PAMM yields the largest max-plus combination of the stored vectors which is less than or equal to the input. Dually, the vector recalled by the min-plus PAMM corresponds to the smallest min-plus combination which is larger than or equal to the input. Apart from unlimited absolute storage capacity and one step retrieval, PAMMs and their compositions exhibit an excellent noise tolerance. Furthermore, the new memories yielded quite promising results in classification problems with a large number of features and classes.

On the Dynamics of Hopfield Neural Networks on Unit Quaternions.

In this paper, we first address the dynamics of the elegant multivalued quaternionic Hopfield neural network (MV-QHNN) proposed by Minemoto et al. Contrary to what was expected, we show that the MV-QHNN, as well as one of its variation, does not always come to rest at an equilibrium state under the usual conditions. In fact, we provide simple examples in which the network yields a periodic sequence of quaternionic state vectors. Afterward, we turn our attention to the continuous-valued quaternionic Hopfield neural network (CV-QHNN), which can be derived from the MV-QHNN by means of a limit process. The CV-QHNN can be implemented more easily than the MV-QHNN model. Furthermore, the asynchronous CV-QHNN always settles down into an equilibrium state under the usual conditions. Theoretical issues are all illustrated by examples in this paper.

Application of self-organising maps towards segmentation of soybean samples by determination of inorganic compounds content.

BACKGROUND: In this study, 20 samples of soybean, both transgenic and conventional cultivars, which were planted in two different regions, Londrina and Ponta Grossa, both located at Paraná, Brazil, were analysed. In order to verify whether the inorganic compound levels in soybeans varied with the region of planting, K, P, Ca, Mg, S, Zn, Mn, Fe, Cu and B contents were analysed by an artificial neural network self-organising map. RESULTS: It was observed that with a topology 10 × 10, 8000 epochs, initial learning rate of 0.1 and initial neighbourhood ratio of 4.5, the network was able to differentiate samples according to region of origin. Among all of the variables analysed by the artificial neural network, the elements Zn, Ca and Mn were those which most contributed to the classification of the samples. CONCLUSION: The results indicated that samples planted in these two regions differ in their mineral content; however, conventional and transgenic samples grown in the same region show no difference in mineral contents in the grain.

Complex-Valued Recurrent Correlation Neural Networks.

In this paper, we generalize the bipolar recurrent correlation neural networks (RCNNs) of Chiueh and Goodman for patterns whose components are in the complex unit circle. The novel networks, referred to as complex-valued RCNNs (CV-RCNNs), are characterized by a possible nonlinear function, which is applied on the real part of the scalar product of the current state and the original patterns. We show that the CV-RCNNs always converge to a stationary state. Thus, they have potential application as associative memories. In this context, we provide sufficient conditions for the retrieval of a memorized vector. Furthermore, computational experiments concerning the reconstruction of corrupted grayscale images reveal that certain CV-RCNNs exhibit an excellent noise tolerance.

Storage and recall capabilities of fuzzy morphological associative memories with adjunction-based learning.

We recently employed concepts of mathematical morphology to introduce fuzzy morphological associative memories (FMAMs), a broad class of fuzzy associative memories (FAMs). We observed that many well-known FAM models can be classified as belonging to the class of FMAMs. Moreover, we developed a general learning strategy for FMAMs using the concept of adjunction of mathematical morphology. In this paper, we describe the properties of FMAMs with adjunction-based learning. In particular, we characterize the recall phase of these models. Furthermore, we prove several theorems concerning the storage capacity, noise tolerance, fixed points, and convergence of auto-associative FMAMs. These theorems are corroborated by experimental results concerning the reconstruction of noisy images. Finally, we successfully employ FMAMs with adjunction-based learning in order to implement fuzzy rule-based systems in an application to a time-series prediction problem in industry.

A class of sparsely connected autoassociative morphological memories for large color images.

This brief introduces a new class of sparsely connected autoassociative morphological memories (AMMs) that can be effectively used to process large multivalued patterns, which include color images as a particular case. Such as the single-valued AMMs, the multivalued models exhibit optimal absolute storage capacity and one-step convergence. The remarkable feature of the proposed models is their sparse structure. In fact, the number of synaptic junctions--and consequently the required computational resources--usually decreases considerably as more and more patterns are stored in the novel multivalued AMMs.

Gray-scale morphological associative memories.

Neural models of associative memories are usually concerned with the storage and the retrieval of binary or bipolar patterns. Thus far, the emphasis in research on morphological associative memory systems has been on binary models, although a number of notable features of autoassociative morphological memories (AMMs) such as optimal absolute storage capacity and one-step convergence have been shown to hold in the general, gray-scale setting. In this paper, we make extensive use of minimax algebra to analyze gray-scale autoassociative morphological memories. Specifically, we provide a complete characterization of the fixed points and basins of attractions which allows us to describe the storage and recall mechanisms of gray-scale AMMs. Computer simulations using gray-scale images illustrate our rigorous mathematical results on the storage capacity and the noise tolerance of gray-scale morphological associative memories (MAMs). Finally, we introduce a modified gray-scale AMM model that yields a fixed point which is closest to the input pattern with respect to the Chebyshev distance and show how gray-scale AMMs can be used as classifiers.