A Novel Fractional Accumulative Grey Model with GA-PSO Optimizer and Its Application.

The prediction of cyber security situation plays an important role in early warning against cyber security attacks. The first-order accumulative grey model has achieved remarkable results in many prediction scenarios. Since recent events have a greater impact on future decisions, new information should be given more weight. The disadvantage of first-order accumulative grey models is that with the first-order accumulative method, equal weight is given to the original data. In this paper, a fractional-order cumulative grey model (FAGM) is used to establish the prediction model, and an intelligent optimization algorithm known as particle swarm optimization (PSO) combined with a genetic algorithm (GA) is used to determine the optimal order. The model discussed in this paper is used for the prediction of Internet cyber security situations. The results of a comparison with the traditional grey model GM(1,1), the grey model GM(1,n), and the fractional discrete grey seasonal model FDGSM(1,1) show that our model is suitable for cases with insufficient data and irregular sample sizes, and the prediction accuracy and stability of the model are better than those of the other three models.

A fractional gradient descent algorithm robust to the initial weights of multilayer perceptron.

For multilayer perceptron (MLP), the initial weights will significantly influence its performance. Based on the enhanced fractional derivative extend from convex optimization, this paper proposes a fractional gradient descent (RFGD) algorithm robust to the initial weights of MLP. We analyze the effectiveness of the RFGD algorithm. The convergence of the RFGD algorithm is also analyzed. The computational complexity of the RFGD algorithm is generally larger than that of the gradient descent (GD) algorithm but smaller than that of the Adam, Padam, AdaBelief, and AdaDiff algorithms. Numerical experiments show that the RFGD algorithm has strong robustness to the order of fractional calculus which is the only added parameter compared to the GD algorithm. More importantly, compared to the GD, Adam, Padam, AdaBelief, and AdaDiff algorithms, the experimental results show that the RFGD algorithm has the best robust performance for the initial weights of MLP. Meanwhile, the correctness of the theoretical analysis is verified.

Principle and Application of Frequency-Domain Characteristic Analysis of Fractional-Order Memristor.

Scaling fractional-order memristor circuit is important for realizing a fractional-order memristor. However, the effective operating-frequency range, operation order, and fractional-order memristance of the scaling fractional-order memristor circuit have not been studied thoroughly; that is, the fractional-order memristance in the effective operating-frequency range has not been calculated quantitatively. The fractional-order memristance is a similar and equally important concept as memristance, memcapacitance, and meminductance. In this paper, the frequency-domain characteristic-analysis principle of the fractional-order memristor is proposed based on the order- and F-frequency characteristic functions. The reasons for selecting the order- and F-frequency characteristic functions are explained. Subsequently, the correctness of the frequency-domain characteristic analysis using the order- and F-frequency characteristic functions is verified from multiple perspectives. Finally, the principle of the frequency-domain characteristic analysis is applied to the recently realized chain-scaling fractional-order memristor circuit. The results of this study indicate that the principle of the frequency-domain characteristic analysis of the fractional-order memristor can successfully calculate the fractional-order memristance of the chain-scaling fractional-order memristor circuit. The proposed principle of frequency-domain characteristic analysis can also be applied to mem-elements, such as memristors, memcapacitors, and meminductors. The main contribution of this study is the principle of the frequency-domain characteristic analysis of the fractional-order memristor based on the order- and F-frequency characteristic functions.

A global neural network learning machine: Coupled integer and fractional calculus operator with an adaptive learning scheme.

Find the global optimal solution of the model is one promising research topic in computational intelligent community. Dependent on analogies to natural processes, the evolutionary swarm intelligent algorithms are widely used for solving global optimization problems which directed by the fitness values. In this paper, we propose one efficient fractional global learning machine (Fragmachine) which includes two stages (descending and ascending) to determine the optimal search path. The neural network model is used to approach the given fitness value. Specifically, for the descending stage, the integer gradient of the network output with respect the current location is employed to find the next descending point, while for the ascending stage, the fractional gradient is implemented to climb and escape from the local optimal point. We further propose one adaptive learning rate during training which relies on both the current gradient (integer or fractional) information and the fitness value. Finally, a series of numerical experiments verify the effectiveness of the proposed algorithm, Fragmachine.

Learning Image Profile Enhancement and Denoising Statistics Priors for Single-Image Super-Resolution.

Single-image super-resolution (SR) has been widely used in computer vision applications. The reconstruction-based SR methods are mainly based on certain prior terms to regularize the SR problem. However, it is very challenging to further improve the SR performance by the conventional design of explicit prior terms. Because of the powerful learning ability, deep convolutional neural networks (CNNs) have been widely used in single-image SR task. However, it is difficult to achieve further improvement by only designing the network architecture. In addition, most existing deep CNN-based SR methods learn a nonlinear mapping function to directly map low-resolution (LR) images to desirable high-resolution (HR) images, ignoring the observation models of input images. Inspired by the split Bregman iteration (SBI) algorithm, which is a powerful technique for solving the constrained optimization problems, the original SR problem is divided into two subproblems: 1) inversion subproblem and 2) denoising subproblem. Since the inversion subproblem can be regarded as an inversion step to reconstruct an intermediate HR image with sharper edges and finer structures, we propose to use deep CNN to capture low-level explicit image profile enhancement prior (PEP). Since the denoising subproblem aims to remove the noise in the intermediate image, we adopt a simple and effective denoising network to learn implicit image denoising statistics prior (DSP). Furthermore, the penalty parameter in SBI is adaptively tuned during the iterations for better performance. Finally, we also prove the convergence of our method. Thus, the deep CNNs are exploited to capture both implicit and explicit image statistics priors. Due to SBI, the SR observation model is also leveraged. Consequently, it bridges between two popular SR approaches: 1) learning-based method and 2) reconstruction-based method. Experimental results show that the proposed method achieves the state-of-the-art SR results.

Feature Selection Using a Neural Network With Group Lasso Regularization and Controlled Redundancy.

We propose a neural network-based feature selection (FS) scheme that can control the level of redundancy in the selected features by integrating two penalties into a single objective function. The Group Lasso penalty aims to produce sparsity in features in a grouped manner. The redundancy-control penalty, which is defined based on a measure of dependence among features, is utilized to control the level of redundancy among the selected features. Both the penalty terms involve the L2,1 -norm of weight matrix between the input and hidden layers. These penalty terms are nonsmooth at the origin, and hence, one simple but efficient smoothing technique is employed to overcome this issue. The monotonicity and convergence of the proposed algorithm are specified and proved under suitable assumptions. Then, extensive experiments are conducted on both artificial and real data sets. Empirical results explicitly demonstrate the ability of the proposed FS scheme and its effectiveness in controlling redundancy. The empirical simulations are observed to be consistent with the theoretical results.

Fractional-order quantum particle swarm optimization.

Motivated by the concepts of quantum mechanics and particle swarm optimization (PSO), quantum-behaved particle swarm optimization (QPSO) was developed to achieve better global search ability. This paper proposes a new method to improve the global search ability of QPSO with fractional calculus (FC). Based on one of the most frequently used fractional differential definitions, the Grünwald-Letnikov definition, we introduce its discrete expression into the position updating of QPSO. Extensive experiments on well-known benchmark functions were performed to evaluate the performance of the proposed fractional-order quantum particle swarm optimization (FQPSO). The experimental results demonstrate its superior ability in achieving optimal solutions for several different optimizations.

Enhanced Non-Local Total Variation Model and Multi-Directional Feature Prediction Prior for Single Image Super Resolution.

It is widely acknowledged that single image super-resolution (SISR) methods play a critical role in recovering the missing high-frequencies in an input low-resolution image. As SISR is severely ill-conditioned, image priors are necessary to regularize the solution spaces and generate the corresponding high-resolution image. In this paper, we propose an effective SISR framework based on the enhanced non-local similarity modeling and learning-based multi-directional feature prediction (ENLTV-MDFP). Since both the modeled and learned priors are exploited, the proposed ENLTV-MDFP method benefits from the complementary properties of the reconstruction-based and learning-based SISR approaches. Specifically, for the non-local similarity-based modeled prior [enhanced non-local total variation, (ENLTV)], it is characterized via the decaying kernel and stable group similarity reliability schemes. For the learned prior [multi-directional feature prediction prior, (MDFP)], it is learned via the deep convolutional neural network. The modeled prior performs well in enhancing edges and suppressing visual artifacts, while the learned prior is effective in hallucinating details from external images. Combining these two complementary priors in the MAP framework, a combined SR cost function is proposed. Finally, the combined SR problem is solved via the split Bregman iteration algorithm. Based on the extensive experiments, the proposed ENLTV-MDFP method outperforms many state-of-the-art algorithms visually and quantitatively.

Fractional-Order Deep Backpropagation Neural Network.

In recent years, the research of artificial neural networks based on fractional calculus has attracted much attention. In this paper, we proposed a fractional-order deep backpropagation (BP) neural network model with L2 regularization. The proposed network was optimized by the fractional gradient descent method with Caputo derivative. We also illustrated the necessary conditions for the convergence of the proposed network. The influence of L2 regularization on the convergence was analyzed with the fractional-order variational method. The experiments have been performed on the MNIST dataset to demonstrate that the proposed network was deterministically convergent and can effectively avoid overfitting.

Semisupervised Tripled Dictionary Learning for Standard-Dose PET Image Prediction Using Low-Dose PET and Multimodal MRI.

OBJECTIVE: To obtain high-quality positron emission tomography (PET) image with low-dose tracer injection, this study attempts to predict the standard-dose PET (S-PET) image from both its low-dose PET (L-PET) counterpart and corresponding magnetic resonance imaging (MRI). METHODS: It was achieved by patch-based sparse representation (SR), using the training samples with a complete set of MRI, L-PET and S-PET modalities for dictionary construction. However, the number of training samples with complete modalities is often limited. In practice, many samples generally have incomplete modalities (i.e., with one or two missing modalities) that thus cannot be used in the prediction process. In light of this, we develop a semisupervised tripled dictionary learning (SSTDL) method for S-PET image prediction, which can utilize not only the samples with complete modalities (called complete samples) but also the samples with incomplete modalities (called incomplete samples), to take advantage of the large number of available training samples and thus further improve the prediction performance. RESULTS: Validation was done on a real human brain dataset consisting of 18 subjects, and the results show that our method is superior to the SR and other baseline methods. CONCLUSION: This paper proposed a new S-PET prediction method, which can significantly improve the PET image quality with low-dose injection. SIGNIFICANCE: The proposed method is favorable in clinical application since it can decrease the potential radiation risk for patients.

Fractional Hopfield Neural Networks: Fractional Dynamic Associative Recurrent Neural Networks.

This paper mainly discusses a novel conceptual framework: fractional Hopfield neural networks (FHNN). As is commonly known, fractional calculus has been incorporated into artificial neural networks, mainly because of its long-term memory and nonlocality. Some researchers have made interesting attempts at fractional neural networks and gained competitive advantages over integer-order neural networks. Therefore, it is naturally makes one ponder how to generalize the first-order Hopfield neural networks to the fractional-order ones, and how to implement FHNN by means of fractional calculus. We propose to introduce a novel mathematical method: fractional calculus to implement FHNN. First, we implement fractor in the form of an analog circuit. Second, we implement FHNN by utilizing fractor and the fractional steepest descent approach, construct its Lyapunov function, and further analyze its attractors. Third, we perform experiments to analyze the stability and convergence of FHNN, and further discuss its applications to the defense against chip cloning attacks for anticounterfeiting. The main contribution of our work is to propose FHNN in the form of an analog circuit by utilizing a fractor and the fractional steepest descent approach, construct its Lyapunov function, prove its Lyapunov stability, analyze its attractors, and apply FHNN to the defense against chip cloning attacks for anticounterfeiting. A significant advantage of FHNN is that its attractors essentially relate to the neuron's fractional order. FHNN possesses the fractional-order-stability and fractional-order-sensitivity characteristics.

Defense Against Chip Cloning Attacks Based on Fractional Hopfield Neural Networks.

This paper presents a state-of-the-art application of fractional hopfield neural networks (FHNNs) to defend against chip cloning attacks, and provides insight into the reason that the proposed method is superior to physically unclonable functions (PUFs). In the past decade, PUFs have been evolving as one of the best types of hardware security. However, the development of the PUFs has been somewhat limited by its implementation cost, its temperature variation effect, its electromagnetic interference effect, the amount of entropy in it, etc. Therefore, it is imperative to discover, through promising mathematical methods and physical modules, some novel mechanisms to overcome the aforementioned weaknesses of the PUFs. Motivated by this need, in this paper, we propose applying the FHNNs to defend against chip cloning attacks. At first, we implement the arbitrary-order fractor of a FHNN. Secondly, we describe the implementation cost of the FHNNs. Thirdly, we propose the achievement of the constant-order performance of a FHNN when ambient temperature varies. Fourthly, we analyze the electrical performance stability of the FHNNs under electromagnetic disturbance conditions. Fifthly, we study the amount of entropy of the FHNNs. Lastly, we perform experiments to analyze the pass-band width of the fractor of an arbitrary-order FHNN and the defense against chip cloning attacks capability of the FHNNs. In particular, the capabilities of defense against chip cloning attacks, anti-electromagnetic interference, and anti-temperature variation of a FHNN are illustrated experimentally in detail. Some significant advantages of the FHNNs are that their implementation cost is considerably lower than that of the PUFs, their electrical performance is much more stable than that of the PUFs under different temperature conditions, their electrical performance stability of the FHNNs under electromagnetic disturbance conditions is much more robust than that of the PUFs, and their amount of entropy is significantly higher than that of the PUFs with the same rank circuit scale.

Statistical iterative reconstruction using adaptive fractional order regularization.

In order to reduce the radiation dose of the X-ray computed tomography (CT), low-dose CT has drawn much attention in both clinical and industrial fields. A fractional order model based on statistical iterative reconstruction framework was proposed in this study. To further enhance the performance of the proposed model, an adaptive order selection strategy, determining the fractional order pixel-by-pixel, was given. Experiments, including numerical and clinical cases, illustrated better results than several existing methods, especially, in structure and texture preservation.

A Random Algorithm for Low-Rank Decomposition of Large-Scale Matrices With Missing Entries.

A random submatrix method (RSM) is proposed to calculate the low-rank decomposition U(m×r)V(n×r)(T) (r < m, n) of the matrix Y∈R(m×n) (assuming m > n generally) with known entry percentage 0 < ρ ≤ 1. RSM is very fast as only O(mr(2)ρ(r)) or O(n(3)ρ(3r)) floating-point operations (flops) are required, compared favorably with O(mnr+r(2)(m+n)) flops required by the state-of-the-art algorithms. Meanwhile, RSM has the advantage of a small memory requirement as only max(n(2),mr+nr) real values need to be saved. With the assumption that known entries are uniformly distributed in Y, submatrices formed by known entries are randomly selected from Y with statistical size k×nρ(k) or mρ(l)×l , where k or l takes r+1 usually. We propose and prove a theorem, under random noises the probability that the subspace associated with a smaller singular value will turn into the space associated to anyone of the r largest singular values is smaller. Based on the theorem, the nρ(k)-k null vectors or the l-r right singular vectors associated with the minor singular values are calculated for each submatrix. The vectors ought to be the null vectors of the submatrix formed by the chosen nρ(k) or l columns of the ground truth of V(T). If enough submatrices are randomly chosen, V and U can be estimated accordingly. The experimental results on random synthetic matrices with sizes such as 13 1072 ×10(24) and on real data sets such as dinosaur indicate that RSM is 4.30 ∼ 197.95 times faster than the state-of-the-art algorithms. It, meanwhile, has considerable high precision achieving or approximating to the best.

Fractional extreme value adaptive training method: fractional steepest descent approach.

The application of fractional calculus to signal processing and adaptive learning is an emerging area of research. A novel fractional adaptive learning approach that utilizes fractional calculus is presented in this paper. In particular, a fractional steepest descent approach is proposed. A fractional quadratic energy norm is studied, and the stability and convergence of our proposed method are analyzed in detail. The fractional steepest descent approach is implemented numerically and its stability is analyzed experimentally.

Using gene expression programming to infer gene regulatory networks from time-series data.

Gene regulatory networks inference is currently a topic under heavy research in the systems biology field. In this paper, gene regulatory networks are inferred via evolutionary model based on time-series microarray data. A non-linear differential equation model is adopted. Gene expression programming (GEP) is applied to identify the structure of the model and least mean square (LMS) is used to optimize the parameters in ordinary differential equations (ODEs). The proposed work has been first verified by synthetic data with noise-free and noisy time-series data, respectively, and then its effectiveness is confirmed by three real time-series expression datasets. Finally, a gene regulatory network was constructed with 12 Yeast genes. Experimental results demonstrate that our model can improve the prediction accuracy of microarray time-series data effectively.

Improved DCT-based nonlocal means filter for MR images denoising.

The nonlocal means (NLM) filter has been proven to be an efficient feature-preserved denoising method and can be applied to remove noise in the magnetic resonance (MR) images. To suppress noise more efficiently, we present a novel NLM filter based on the discrete cosine transform (DCT). Instead of computing similarity weights using the gray level information directly, the proposed method calculates similarity weights in the DCT subspace of neighborhood. Due to promising characteristics of DCT, such as low data correlation and high energy compaction, the proposed filter is naturally endowed with more accurate estimation of weights thus enhances denoising effectively. The performance of the proposed filter is evaluated qualitatively and quantitatively together with two other NLM filters, namely, the original NLM filter and the unbiased NLM (UNLM) filter. Experimental results demonstrate that the proposed filter achieves better denoising performance in MRI compared to the others.

Efficient CT metal artifact reduction based on fractional-order curvature diffusion.

We propose a novel metal artifact reduction method based on a fractional-order curvature driven diffusion model for X-ray computed tomography. Our method treats projection data with metal regions as a damaged image and uses the fractional-order curvature-driven diffusion model to recover the lost information caused by the metal region. The numerical scheme for our method is also analyzed. We use the peak signal-to-noise ratio as a reference measure. The simulation results demonstrate that our method achieves better performance than existing projection interpolation methods, including linear interpolation and total variation.

A new CT metal artifacts reduction algorithm based on fractional-order sinogram inpainting.

In this paper, we propose a new metal artifacts reduction algorithm based on fractional-order total-variation sinogram inpainting model for X-ray computed tomography (CT). The numerical algorithm for our fractional-order framework is also analyzed. Simulations show that, both quantitatively and qualitatively, our method is superior to conditional interpolation methods and the classic integral-order total variation model.

Fractional differential mask: a fractional differential-based approach for multiscale texture enhancement.

In this paper, we intend to implement a class of fractional differential masks with high-precision. Thanks to two commonly used definitions of fractional differential for what are known as GrUmwald-Letnikov and Riemann-Liouville, we propose six fractional differential masks and present the structures and parameters of each mask respectively on the direction of negative x-coordinate, positive x-coordinate, negative y-coordinate, positive y-coordinate, left downward diagonal, left upward diagonal, right downward diagonal, and right upward diagonal. Moreover, by theoretical and experimental analyzing, we demonstrate the second is the best performance fractional differential mask of the proposed six ones. Finally, we discuss further the capability of multiscale fractional differential masks for texture enhancement. Experiments show that, for rich-grained digital image, the capability of nonlinearly enhancing complex texture details in smooth area by fractional differential-based approach appears obvious better than by traditional intergral-based algorithms.