Complex-valued Associative Memory Research Articles

Projection rule for complex-valued associative memory with large constant terms

Projection rule for complex-valued associative memory with large constant terms

Gradient Descent Learning for Rotor Associative Memory

Complex-valued Associative Memory (CAM) is an extended model of Hopfield Associative Memory (HAM). The fundamental elements, such as input-output signals and connection weights of the CAM are extended to complex numbers. The CAM can deal with multi-states information. Rotor Associative Memory (RAM) is an extended model of the CAM. Rotor neurons are essentially equivalent to complex-valued neurons. Connection weights of the RAM are expressed by two by two matrices. Only hebb rule has been proposed for the learning of the RAM. Its storage capacity is small, so advanced learning methods are necessary. In this paper, we propose gradient descent learning rule for the RAM (GDR RAM). It is based on that for the CAM (GDR CAM) proposed by Lee. We solved the learning rule and performed computer simulations to compare the GDR CAM and the GDR RAM. At last, it turned out that the storage capacity of the GDR RAM is approximately twice as much as that of the GDR CAM and the noise robustness of the GDR RAM is much better than that of the GDR CAM.

Noise Robust Gradient Descent Learning for Complex-Valued Associative Memory

Complex-valued Associative Memory (CAM) is an advanced model of Hopfield Associative Memory. The CAM is based on multi-state neurons and has the high ability of representation. Lee proposed gradient descent learning for the CAM to improve the storage capacity. It is based on only the phases of input signals. In this paper, we propose another type of gradient descent learning based on both the phases and the amplitude. The proposed learning method improves the noise robustness and accelerates the learning speed.

Reducing Spurious States by Rotor Associative Memory

Hopfield Associative Memory (HAM) is a typical model of associative memories. Complex-valued Associative Memory is a complex-valued version of HAM. They store not only learning patterns but other undesired patterns, which are called spurious states. The reversed patterns in HAM and the rotated patterns in CAM are typical spurious states. We propose to reduce spurious states by using Rotor Associative Memory (RAM). The RAM does not store the rotated patterns except the reversed patterns. So the RAM removes most of spurious states from the CAM. By adding a dummy state, the RAM also can remove them from the HAM. To find spurious states, we construct a searching system similar to the chaotic CAM. By using the searching system, we investigate whether the RAM can reduce the number of spurious states.

PSEUDO-RELAXATION LEARNING ALGORITHM FOR COMPLEX-VALUED ASSOCIATIVE MEMORY

HAM (Hopfield Associative Memory) and BAM (Bidirectinal Associative Memory) are representative associative memories by neural networks. The storage capacity by the Hebb rule, which is often used, is extremely low. In order to improve it, some learning methods, for example, pseudo-inverse matrix learning and gradient descent learning, have been introduced. Oh introduced pseudo-relaxation learning algorithm to HAM and BAM. In order to accelerate it, Hattori proposed quick learning. Noest proposed CAM (Complex-valued Associative Memory), which is complex-valued HAM. The storage capacity of CAM by the Hebb rule is also extremely low. Pseudo-inverse matrix learning and gradient descent learning have already been generalized to CAM. In this paper, we apply pseudo-relaxation learning algorithm to CAM in order to improve the capacity.

Complex-valued Multidirectional Associative Memory

Hopfield model is a representative associative memory. It was improved to Bidirectional Associative Memory(BAM) by Kosko and Multidirectional Associative Memory(MAM) by Hagiwara. They have two layers or multilayers. Since they have symmetric connections between layers, they ensure to converge. MAM can deal with multiples of many patterns, such as (x1, x2,…), where xm is the pattern on layer-m. Noest, Hirose and Nemoto proposed complex-valued Hopfield model. Lee proposed complex-valued Bidirectional Associative Memory. Zemel proved the rotation invariance of complex-valued Hopfield model. It means that the rotated pattern also stored.In this paper, the complex-valued Multidirectional Associative Memory is proposed. The rotation invariance is also proved. Moreover it is shown by computer simulation that the differences of angles of given patterns are automatically reduced.At first we define complex-valued Multidirectional Associative Memory. Then we define the energy function of network. By using energy function, we prove that the network ensures to converge.Next, we define the learning law and show the characteristic of recall process. The characteristic means that the differences of angles of given patterns are automatically reduced. Especially we prove the following theorem. In case that only a multiple of patterns is stored, if patterns with different angles are given to each layer, the differences are automatically reduced.Finally, we invest that the differences of angles influence the noise robustness. It reduce the noise robustness, because input to each layer become small. We show that by computer simulations.

Characteristics of the complex-valued associative memory model having penalty term

Characteristics of the complex-valued associative memory model having penalty term

Characteristics of the complex-valued associative memory model having penalty term

In the complex-valued associative memory model in which each variable of the network has a complex value, a model is proposed in which a penalty term is introduced into the energy function. In the model treated in this paper, some rules are set up in the fabrication of the memorized pattern, and these rules are reflected in the energy function as a penalty term. In the case of recall of memory, a rule is used as empirical information so that the memory capacity and recall performance are improved. As an example of fabricating the memorization pattern, the entire pattern is divided into two and the neurons corresponding to both satisfy the complex conjugate relationship. As an application of the complex-valued associative memory model, the use of the communication code in the error correcting system can be considered. By comparison with the typical conventional communication system, the new model is shown to exhibit excellent performance. © 2000 Scripta Technica, Electron Comm Jpn Pt 3, 83(7): 62–70, 2000

A complex-valued associative memory for storing patterns as oscillatory states

A neuron model in which the neuron state is described by a complex number is proposed. A network of these neurons, which can be used as an associative memory, operates in two distinct modes: (i) fixed point mode and (ii) oscillatory mode. Mode selection can be done by varying a continuous mode parameter, \(\nu\), between \(0\) and \(1\). At one extreme value of \(\nu\) (\(=0\)), the network has conservative dynamics, and at the other (\(\nu = 1\)), the dynamics are dissipative and governed by a Lyapunov function. Patterns can be stored and retrieved at any value of \(\nu\) by, (i) a one-step outer product rule or (ii) adaptive Hebbian learning. In the fixed point mode patterns are stored as fixed points, whereas in the oscillatory mode they are encoded as phase relations among individual oscillations. By virtue of an instability in the oscillatory mode, the retrieval pattern is stable over a finite interval, the stability interval, and the pattern gradually deteriorates with time beyond this interval. However, at certain values of \(\nu\) sparsely distributed over \(\nu\)-space the instability disappears. The neurophysiological significance of the instability is briefly discussed. The possibility of physically interpreting dissipativity and conservativity is explored by noting that while conservativity leads to energy savings, dissipativity leads to stability and reliable retrieval.