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Diversity is supposed to create better groups and societies but sometimes fails. It is explained why the power of diversity may not create better groups in the current diversity prediction theory. Diversity may hurt civic life and introduce distrust. This is because the current diversity prediction theory is based on real numbers that ignore individual abilities. Its diversity prediction theory maximizes performance with infinite population size. Contrary to this, collective intelligence or swarm intelligence is not maximized by infinite population size, but by population size. The extended diversity prediction theory using the complex number allows us to express individual abilities or qualities. The diversity of complex numbers always produces better groups and societies. The wisdom of crowds, collective intelligence, swarm intelligence or nature-inspired intelligence is implemented in the current machine learning or artificial intelligence, called Random Forest. The problem of the current diversity prediction theory is detailed in this paper. • Diversity is supposed to create better groups and societies but sometimes fails. • Diversity in real numbers ignores individual abilities. • Diversity of population size plays a key role in collective intelligence. • Diversity in complex numbers allows us to express individual abilities. • Diversity in complex numbers always produces better groups and societies. [ABSTRACT FROM AUTHOR]

A complex-valued Hopfield neural network (CHNN) is a model of a multistate Hopfield neural network, and has been applied to the storage of multilevel data. Weak noise tolerance, however, is a disadvantage of CHNNs. Symmetric CHNNs (SCHNNs), modified CHNNs, improve the noise tolerance of CHNNs. In the present work, we study the global and local minima of SCHNNs with one training pattern. In CHNNs, the global minima are the training and rotated patterns, and there are no local minima. In SCHNNs, it has been hard to determine all the global and local minima. It is thought that the global minima are the training and reversed patterns, and that there are no local minima in most cases. In the present work, however, we find many local minima, and show that they are very weak attractors, which reduce noise tolerance very little. In addition, we determine all the global minima of SCHNNs. [ABSTRACT FROM AUTHOR]

• The impact of loss functions on complex MRI reconstruction with CNNs was analyzed. • The mean squared loss is biased towards magnitude underestimations with phase error. • A newly-proposed unbiased loss function called ⊥ -loss yields improved performance. • ⊥ -loss can be applied to any high-dimensional problem defined in R 2. • ⊥ -loss was applied to image registration tasks, showing superior performance. Convolutional neural networks (CNNs) are increasingly adopted in medical imaging, e.g., to reconstruct high-quality images from undersampled magnetic resonance imaging (MRI) acquisitions or estimate subject motion during an examination. MRI is naturally acquired in the complex domain C , obtaining magnitude and phase information in k-space. However, CNNs in complex regression tasks are almost exclusively trained to minimize the L2 loss or maximizing the magnitude structural similarity (SSIM), which are possibly not optimal as they do not take full advantage of the magnitude and phase information present in the complex domain. This work identifies that minimizing the L2 loss in the complex field has an asymmetry in the magnitude/phase loss landscape and is biased, underestimating the reconstructed magnitude. To resolve this, we propose a new loss function for regression in the complex domain called ⊥ -loss, which adds a novel phase term to established magnitude loss functions, e.g., L2 or SSIM. We show ⊥ -loss is symmetric in the magnitude/phase domain and has favourable properties when applied to regression in the complex domain. Specifically, we evaluate the ⊥ + ℓ 2 -loss and ⊥ +SSIM-loss for complex undersampled MR image reconstruction tasks and MR image registration tasks. We show that training a model to minimize the ⊥ + ℓ 2 -loss outperforms models trained to minimize the L2 loss and results in similar performance compared to models trained to maximize the magnitude SSIM while offering high-quality phase reconstruction. Moreover, ⊥ -loss is defined in R n , and we apply the loss function to the R 2 domain by learning 2D deformation vector fields for image registration. We show that a model trained to minimize the ⊥ + ℓ 2 -loss outperforms models trained to minimize the end-point error loss. [ABSTRACT FROM AUTHOR]

Abstract: A family of new complex number chaotic maps based image encryption algorithm is proposed in the paper. A family of maps is constructed and proved to be chaotic in the complex number field, and its characteristics are analyzed. Two maps are selected from the chaotic maps family and are utilized to construct pseudorandom keystream sequence. In the proposed encryption algorithm, the pseudorandom keystream sequences are used to scramble and diffuse the plain image data and two entropy coding methods are used to reduce the correlation among the signals. Both theoretical analysis and experimental tests show that the proposed algorithm is secure and efficient. [Copyright &y& Elsevier]

Abstract: In the existing literature, the information regarding the exact solutions for free in-plane vibrations of the curved beams (or arches) carrying various concentrated elements is rare, particularly for the case with multiple attachments including eccentricities and mass moments of inertias. For this reason, this paper aims at presenting an effective approach to tackle the title problem. First of all, the un-coupled equation of motion for the circumferential displacement of an arch segment is derived. Next, based on the value of the discriminate parameter for a cubic equation, the exact solutions for the three types of roots of the un-coupled equation are determined and, corresponding to each type of roots, all displacement functions for the arch segment in terms of the real numbers (instead of the complex ones) are obtained. Finally, use of the compatible equations for the displacements and slopes together with the equilibrium equations for the forces and moments at each intermediate node and two ends of the entire curved beam, a frequency equation of the form ∣H(ω)∣=0 is obtained. It is found that the conventional approach by using the condition “∣H(ω t )∣⩽ ε” to search for the approximate value of ω t is difficult even if the convergence tolerance ε is greater than 10+3 (i.e., ε >10+3) instead of less than 10−3 (i.e., ε <10−3), however, the half-interval method is one of the effective tools for solving the problem if all coefficients of the determinant ∣H(ω)∣ are the real numbers. In addition to comparing with the existing literature, most of the numerical results obtained from the presented method are compared with those obtained from the conventional finite element method (FEM) and good agreement is achieved. [Copyright &y& Elsevier]

Abstract: A critical point is a point on which the derivatives of an error function are all zero. It has been shown in the literatures that the critical points caused by the hierarchical structure of the real-valued neural network could be local minima or saddle points, whereas most of the critical points caused by the hierarchical structure are saddle points in the case of complex- valued neural networks. Several studies have demonstrated that that kind of singularity has a negative effect on learning dynamics in neural networks. In this paper, we will demonstrate via some examples that the decomposition of high- dimensional NNs into real-valued NNs equivalent to the original NNs yields the NNs that do not have critical points based on the hierarchical structure. [Copyright &y& Elsevier]

Abstract: Most of local minima caused by the hierarchical structure can be resolved by extending the real-valued neural network to complex numbers. It was proved in 2000 that a critical point of the real-valued neural network with hidden neurons always gives many critical points of the real-valued neural network with hidden neurons. These critical points consist of many lines in the parameter space which could be local minima or saddle points. Local minima cause plateaus which have a strong negative influence on learning. However, most of the critical points of complex-valued neural network are saddle points unlike those of the real-valued neural network. This is a prominent property of the complex-valued neural network. [Copyright &y& Elsevier]

Binarized neural networks, or BNNs, show great promise in edge-side applications with resource limited hardware, but raise the concerns of reduced accuracy. Motivated by the complex neural networks, in this paper we introduce complex representation into the BNNs and propose Binary complex neural network – a novel network design that processes binary complex inputs and weights through complex convolution, but still can harvest the extraordinary computation efficiency of BNNs. To ensure fast convergence rate, we propose novel BCNN based batch normalization and weight initialization strategies. Experimental results on image and radio signal classifications show that BCNN can achieve better accuracy compared to the original BNN models. BCNN improves BNN by strengthening its learning capability through complex representation and extending its applicability to complex-valued input data. Our code is available at https://github.com/flying-Yan/BCNN. [ABSTRACT FROM AUTHOR]

In this letter, we will clarify the redundancy of the parameters of the complex-valued neural network. The results may be summarized as follows. There exist the four transformations which can cause the redundancy of the parameters of the complex-valued neural network, including the two transformations which can cause the redundancy of the parameters of the real-valued neural network (i.e., the interchange of the two hidden neurons and the sign flip of the parameters on the hidden neurons). [Copyright &y& Elsevier]