Frequency-Multiplexing Ability of Complex-Valued Hebbian Learning in Logic Gates

Abstract

Lightwave has attractive characteristics such as spatial parallelism, temporal rapidity in signal processing, and frequency band vastness. In particular, the vast carrier frequency bandwidth promises novel information processing. In this paper, we propose a novel optical logic gate that learns multiple functions at frequencies different from one another, and analyze the frequency-domain multiplexing ability in the learning based on complex-valued Hebbian rule. We evaluate the averaged error function values in the learning process and the error probabilities in the realized logic functions. We investigate optimal learning parameters as well as performance dependence on the number of learning iterations and the number of parallel paths per neuron. Results show a trade-off among the learning parameters such as learning time constant and learning gain. We also find that when we prepare 10 optical path differences and conduct 200 learning iterations, the error probability completely decreases to zero in a three-function multiplexing case. However, at the same time, the error probability is tolerant of the path number. That is, even if the path number is reduced by half, error probability is found almost zero. The results can be useful to determine neural parameters for future optical neural network systems and devices that utilize the vast frequency bandwidth for frequency-domain multiplexing.

Nature of complex number and complex-valued neural networks

“…We can obtain a set of results, AND and XOR, as we intended, with natural generalization characteristics. We can also estimate the realizable logic density in the frequency domain [25]. Figure 4 illustrates that such a FDM learning logic circuit can be equivalent with a number of conventional logic.…”

“…Accordingly, the amplitude-phase nonlinearity is more suitable for wave-related processing. Actually, based on the amplitude-phase nonlinearity, we have proposed new adaptive systems such as the optical learning logic circuits realizing frequency-multiplexed operation [25] and the fast method to yield CGH for threedimensional movies [26,27] reviewed in Sect. 2.…”

“…radars to visualize plastic landmines [10][11][12][13][14][15] and satellite radars to estimate landscape information [16] and/or land-use classification [17], blur-compensation image processing [18], filtering and other time-sequential signal processing [19,20], frequency-domain multiplexed (FDM) microwave signal processing [21] and pulse beamforming in ultra-wideband (UWB) communications [22], FDM neural networks and learning logic circuits using lightwave [4,[23][24][25] and fast adaptive threedimensional holographic movie generation for optical tweezers [26,27], and developmental learning of motion control in combination with reinforcement learning [28]. In parallel, general associative memories [29] and independent component analysis (ICA) neural networks [30 31] are also making progress in their improvement.…”

Applications of Complex-Valued Self-Organizing Maps to Ground Penetrating Radar Imaging Systems

“…This type of multiplication reduces ineffective degree of freedom in the learning or self-organization to enhance the generalization characteristics in comparison with double-dimensional real-number networks, in spite of the fact that a complex number can be represented as an ordered pair of real numbers. The network dynamics consisting of this elemental rotation and amplification / attenuation leads to significant merits in total, originating from the consistency with the wave-related phenomena and information in the GPRs and other coherent systems (Kawata & Hirose, 2008) (Tay et al, 2008). For details, see (Hirose, 2006) (Hirose, n.d.).…”

A Meta-cognitive Fully Complex Valued Functional Link predictor Network for solving real valued prediction problems

“…These signals are required to be represented in the Complex domain in order to preserve their physical uniqueness. Hence, it is necessary to create adaptively learning algorithms in the area of Complex valued neural networks [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19] [20], [21]. For the complete details of the complex-valued neural networks in the literature, one must refer to [22], [23], [24].…”