Deep convolutional neural networks (CNNs) have achieved remarkable results in image processing tasks. However, their high expression ability risks overfitting. Consequently, data augmentation techniques have been proposed to prevent overfitting while enriching datasets. Recent CNN architectures with more parameters are rendering traditional data augmentation techniques insufficient. In this study, we propose a new data augmentation technique called random image cropping and patching (RICAP) which randomly crops four images and patches them to create a new training image. Moreover, RICAP mixes the class labels of the four images, resulting in an advantage of the soft labels. We evaluated RICAP with current state-of-the-art CNNs (e.g., the shake-shake regularization model) by comparison with competitive data augmentation techniques such as cutout and mixup. RICAP achieves a new state-of-the-art test error of 2.19% on CIFAR-10. We also confirmed that deep CNNs with RICAP achieve better results on classification tasks using CIFAR-100 and ImageNet, an image-caption retrieval task using Microsoft COCO, and other computer vision tasks.

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Data Augmentation Using Random Image Cropping and Patching for Deep CNNs

A review of definitions of fractional derivatives and other operators

On fractional calculus with general analytic kernels

Fractional calculus dates its inception to a correspondence between Leibniz and L’Hopital in 1695, when Leibniz described “paradoxes” and predicted that “one day useful consequences will be drawn” from them. In today’s world, the study of non-integer orders of differentiation has become a thriving field of research, not only in mathematics but also in other parts of science such as physics, biology, and engineering: many of the “useful consequences” predicted by Leibniz have been discovered. However, the field has grown so far that researchers cannot yet agree on what a “fractional derivative” can be. In this manuscript, we suggest and justify the idea of classification of fractional calculus into distinct classes of operators.

https://www.mdpi.com/2227-7390/7/9/830/pdf

On Fractional Operators and Their Classifications

The remarkable flexibility and adaptability of ensemble methods and deep learning models have led to the proliferation of their application in bioinformatics research. Traditionally, these two machine learning techniques have largely been treated as independent methodologies in bioinformatics applications. However, the recent emergence of ensemble deep learning—wherein the two machine learning techniques are combined to achieve synergistic improvements in model accuracy, stability and reproducibility—has prompted a new wave of research and application. Here, we share recent key developments in ensemble deep learning and look at how their contribution has benefited a wide range of bioinformatics research from basic sequence analysis to systems biology. While the application of ensemble deep learning in bioinformatics is diverse and multifaceted, we identify and discuss the common challenges and opportunities in the context of bioinformatics research. We hope this Review Article will bring together the broader community of machine learning researchers, bioinformaticians and biologists to foster future research and development in ensemble deep learning, and inspire novel bioinformatics applications that are unattainable by traditional methods. Recent developments in machine learning have seen the merging of ensemble and deep learning techniques. The authors review advances in ensemble deep learning methods and their applications in bioinformatics, and discuss the challenges and opportunities going forward.

Ensemble deep learning in bioinformatics

Fractional calculus is a powerful and effective tool for modelling nonlinear systems. In this paper, we introduce a class of new fractional derivative named general conformable fractional derivative (GCFD) to describe the physical world. The GCFD is generalized from the concept of conformable fractional derivative (CFD) proposed by Khalil. We point out that the term $$t^{1-\alpha }$$
 in CFD definition is not essential and it is only a kind of “fractional conformable function”. We also give physical and geometrical interpretations of GCFD which thus indicate potential applications in physics and engineering. It is easy to demonstrate that CFD is a special case of GCFD, then to the authors’ knowledge, so far we first give the physical and geometrical interpretations of CFD. The above work is done by a new framework named Extended Gâteaux derivative and Linear Extended Gâteaux derivative which are natural extensions of Gâteaux derivative. As an application, we discuss a scheme for solving fractional differential equations of GCFD.

General conformable fractional derivative and its physical interpretation

Representations of acting processes and memory effects: General fractional derivative and its application to theory of heat conduction with finite wave speeds

Some applications of fractional partial differential equations (FPDEs) based on conformable fractional derivative (CFD) are discussed recently. As an extension of CFD — power kernel is generalized to other normalized probability distribution kernels, general conformable fractional derivative (GCFD) provides a framework to explain the physical meaning of CFD and describe which phenomena or processes GCFD/CFD are suitable to model. We will establish the multivariate theory of GCFD in this paper. Compared with traditional hereditary fractional calculus like Riemann–Liouville type or Caputo type, the unified theory of GCFD vector calculus has well forms with simplicity and elegance. As applications, fractional Maxwell’s equations of GCFD are given to describe electromagnetic fields of media that demonstrate inhomogeneous, complicated and local properties, and some fractional parabolic equations are presented as natural extensions of some nonlinear parabolic equations that cover several important famous equations.

A new framework for multivariate general conformable fractional calculus and potential applications

Equivalence between dropout and data augmentation: A mathematical check.

Most disordered dielectrics especially solid dielectrics show non-Debye laws, and many empirical approximation models are proposed to describe these anomalous relaxation processes. Fractional calculus approach is a popular approach to analyze the anomalous relaxation processes and has been intensively studied in various dielectric materials. However, Capelas de Oliveira et al. proved that the memory kernels of Caputo type fractional derivatives must satisfy the initial value condition (IVC). But both kernels of the Caputo–Fabrizio derivative and the Atangana–Baleanu derivative do not satisfy this condition. In this paper, we prove that the Caputo type derivative with a Prabhakar-like kernel satisfies the IVC and this derivative is in the framework of general Caputo fractional derivative (GC derivative). Corresponding anomalous relaxation model and its solution are discussed. Analysis result shows our model, as a direct extension of the Cole–Cole model, contains the Debye model and the fractional relaxation model as particular cases.

Dazhi Zhao

Papers

General conformable fractional derivative and its physical interpretation

Representations of acting processes and memory effects: General fractional derivative and its application to theory of heat conduction with finite wave speeds

A new framework for multivariate general conformable fractional calculus and potential applications

Equivalence between dropout and data augmentation: A mathematical check.

Anomalous relaxation model based on the fractional derivative with a Prabhakar-like kernel