Activation functions play a key role in providing remarkable performance in deep neural networks, and the rectified linear unit (ReLU) is one of the most widely used activation functions. Various new activation functions and improvements on ReLU have been proposed, but each carry performance drawbacks. In this paper, we propose an improved activation function, which we name the natural-logarithm-rectified linear unit (NLReLU). This activation function uses the parametric natural logarithmic transform to improve ReLU and is simply defined as. NLReLU not only retains the sparse activation characteristic of ReLU, but it also alleviates the "dying ReLU" and vanishing gradient problems to some extent. It also reduces the bias shift effect and heteroscedasticity of neuron data distributions among network layers in order to accelerate the learning process. The proposed method was verified across ten convolutional neural networks with different depths for two essential datasets. Experiments illustrate that convolutional neural networks with NLReLU exhibit higher accuracy than those with ReLU, and that NLReLU is comparable to other well-known activation functions. NLReLU provides 0.16% and 2.04% higher classification accuracy on average compared to ReLU when used in shallow convolutional neural networks with the MNIST and CIFAR-10 datasets, respectively. The average accuracy of deep convolutional neural networks with NLReLU is 1.35% higher on average with the CIFAR-10 dataset.
University of Alberta（阿尔伯塔大学）；Hong Kong University of Science and Technology（香港科技大学）；University of Washington（华盛顿大学）；Carnegie Mellon University（卡内基梅隆大学）
In this paper we investigate the performance of different types of rectified activation functions in convolutional neural network: standard rectified linear unit (ReLU), leaky rectified linear unit (Leaky ReLU), parametric rectified linear unit (PReLU) and a new randomized leaky rectified linear units (RReLU). We evaluate these activation function on standard image classification task. Our experiments suggest that incorporating a non-zero slope for negative part in rectified activation units could consistently improve the results. Thus our findings are negative on the common belief that sparsi…
In past few years, linear rectified unit activation functions have shown its significance in the neural networks, surpassing the performance of sigmoid activations. RELU (Nair & Hinton, 2010), ELU (Clevert et al., 2015), PRELU (He et al., 2015), LRELU (Maas et al., 2013), SRELU (Jin et al., 2016), ThresholdedRELU, all these linear rectified activation functions have its own significance over others in some aspect. Most of the time these activation functions suffer from bias shift problem due to non-zero output mean, and high weight update problem in deep complex networks due to unit gradient, …
Johann Radon Institute（约翰·拉东研究所）；University of Vienna（维也纳大学）
In this effort, we derive a formula for the integral representation of a shallow neural network with the Rectified Power Unit activation function. Mainly, our first result deals with the univariate case of representation capability of RePU shallow networks. The multidimensional result in this paper characterizes the set of functions that can be represented with bounded norm and possibly unbounded width.
Element-wise activation functions play a critical role in deep neural networks via affecting the expressivity power and the learning dynamics. Learning-based activation functions have recently gained increasing attention and success. We propose a new perspective of learnable activation function through formulating them with element-wise attention mechanism. In each network layer, we devise an attention module which learns an element-wise, sign-based attention map for the pre-activation feature map. The attention map scales an element based on its sign. Adding the attention module with a rectif…
IDLab, ELIS, Ghent University – imec（根特大学-imec的IDLab和ELIS（根特大学全球校区生物技术数据科学中心））；Center for Biotech Data Science, Ghent University Global Campus（根特大学全球校区生物技术数据科学中心）
In this paper, we introduce a novel type of Rectified Linear Unit (ReLU), called a Dual Rectified Linear Unit (DReLU). A DReLU, which comes with an unbounded positive and negative image, can be used as a drop-in replacement for a tanh activation function in the recurrent step of Quasi-Recurrent Neural Networks (QRNNs) (Bradbury et al. (2017)). Similar to ReLUs, DReLUs are less prone to the vanishing gradient problem, they are noise robust, and they induce sparse activations. We independently reproduce the QRNN experiments of Bradbury et al. (2017) and compare our DReLU-based QRNNs with the ori…
University of Queensland（昆士兰大学）；International Computer Science Institute（国际计算机科学研究所）
An interesting approach to analyzing neural networks that has received renewed attention is to examine the equivalent kernel of the neural network. This is based on the fact that a fully connected feedforward network with one hidden layer, a certain weight distribution, an activation function, and an infinite number of neurons can be viewed as a mapping into a Hilbert space. We derive the equivalent kernels of MLPs with ReLU or Leaky ReLU activations for all rotationally-invariant weight distributions, generalizing a previous result that required Gaussian weight distributions. Additionally, th…
Charles Sturt University（查尔斯特大学）；Asia Pacific International College（亚太国际学院）；Worcester State University（伍斯特州立大学）；University of Kurdistan Hewler（库尔德斯坦埃尔比勒大学）
Background and Aim: Each stage of sleep can affect human health, and not getting enough sleep at any stage may lead to sleep disorder like parasomnia, apnea, insomnia, etc. Sleep-related diseases could be diagnosed using Convolutional Neural Network Classifier. However, this classifier has not been successfully implemented into sleep stage classification systems due to high complexity and low accuracy of classification. The aim of this research is to increase the accuracy and reduce the learning time of Convolutional Neural Network Classifier. Methodology: The proposed system used a modified O…
Activation functions have a notorious impact on neural networks on both training and testing the models against the desired problem. Currently, the most used activation function is the Rectified Linear Unit (ReLU). This paper introduces a new and novel activation function, closely related with the new activation $Swish = x * sigmoid(x)$ (Ramachandran et al., 2017) which generalizes it. We call the new activation $E-swish = βx * sigmoid(x)$. We show that E-swish outperforms many other well-known activations including both ReLU and Swish. For example, using E-swish provided 1.5% and 4.6% accurac…
Université Savoie Mont Blanc（萨瓦大学）；TLECOM Bretagne（布列塔尼电信学院）
The paper proposes representation functionals in a dual paradigm where learning jointly concerns both linear convolutional weights and parametric forms of nonlinear activation functions. The nonlinear forms proposed for performing the functional representation are associated with a new class of parametric neural transfer functions called rectified power sigmoid units. This class is constructed to integrate both advantages of sigmoid and rectified linear unit functions, in addition with rejecting the drawbacks of these functions. Moreover, the analytic form of this new neural class involves sca…
University of California - Irvine（加州大学尔湾分校）；Adobe Research（奥多比研究院）
Artificial neural networks typically have a fixed, non-linear activation function at each neuron. We have designed a novel form of piecewise linear activation function that is learned independently for each neuron using gradient descent. With this adaptive activation function, we are able to improve upon deep neural network architectures composed of static rectified linear units, achieving state-of-the-art performance on CIFAR-10 (7.51%), CIFAR-100 (30.83%), and a benchmark from high-energy physics involving Higgs boson decay modes.
Amazon Web Services（亚马逊云科技）；University of Massachusetts Amherst（马萨诸塞大学阿默斯特分校）；Microsoft Inc.（微软公司）
Most deep neural networks use simple, fixed activation functions, such as sigmoids or rectified linear units, regardless of domain or network structure. We introduce differential equation units (DEUs), an improvement to modern neural networks, which enables each neuron to learn a particular nonlinear activation function from a family of solutions to an ordinary differential equation. Specifically, each neuron may change its functional form during training based on the behavior of the other parts of the network. We show that using neurons with DEU activation functions results in a more compact …
The University of Texas at Austin（德克萨斯大学奥斯汀分校）；Cognizant Technology Solutions（高知特技术解决方案公司）
The choice of activation function can have a large effect on the performance of a neural network. While there have been some attempts to hand-engineer novel activation functions, the Rectified Linear Unit (ReLU) remains the most commonly-used in practice. This paper shows that evolutionary algorithms can discover novel activation functions that outperform ReLU. A tree-based search space of candidate activation functions is defined and explored with mutation, crossover, and exhaustive search. Experiments on training wide residual networks on the CIFAR-10 and CIFAR-100 image datasets show that t…
University of Warwick（华威大学）；Pakistan Institute of Engineering and Applied Sciences (PIEAS)（巴基斯坦工程与应用科学研究院）
An artificial neuron is modelled as a weighted summation followed by an activation function which determines its output. A wide variety of activation functions such as rectified linear units (ReLU), leaky-ReLU, Swish, MISH, etc. have been explored in the literature. In this short paper, we explore what happens when the activation function of each neuron in an artificial neural network is learned natively from data alone. This is achieved by modelling the activation function of each neuron as a small neural network whose weights are shared by all neurons in the original network. We list our pri…
Poisson Hyperplane Processes with Rectified Linear Units
Institute of Mathematical Sciences, ShanghaiTech University, China（上海科技大学数学科学研究所）；Department of Statistics and Actuarial Science, Simon Fraser University, Canada（西蒙弗雷泽大学统计与精算科学系）
Amazon Inc.（亚马逊公司）；University of Oregon（俄勒冈大学）；Microsoft Inc.（微软公司）；University of Massachusetts Amherst（马萨诸塞大学阿默斯特分校）
Most deep neural networks use simple, fixed activation functions, such as sigmoids or rectified linear units, regardless of domain or network structure. We introduce differential equation units (DEUs), an improvement to modern neural networks, which enables each neuron to learn a particular nonlinear activation function from a family of solutions to an ordinary differential equation. Specifically, each neuron may change its functional form during training based on the behavior of the other parts of the network. We show that using neurons with DEU activation functions results in a more compact …
Simons Center for Systems Biology（西蒙斯系统生物学中心）；Institute for Advanced Study（高等研究院）；Princeton Neuroscience Institute（普林斯顿神经科学研究所）；Princeton University（普林斯顿大学）
A model of associative memory is studied, which stores and reliably retrieves many more patterns than the number of neurons in the network. We propose a simple duality between this dense associative memory and neural networks commonly used in deep learning. On the associative memory side of this duality, a family of models that smoothly interpolates between two limiting cases can be constructed. One limit is referred to as the feature-matching mode of pattern recognition, and the other one as the prototype regime. On the deep learning side of the duality, this family corresponds to feedforward…
The University of Texas at Austin（德克萨斯大学奥斯汀分校）；Cognizant AI Labs（高知特人工智能实验室）
Recent studies have shown that the choice of activation function can significantly affect the performance of deep learning networks. However, the benefits of novel activation functions have been inconsistent and task dependent, and therefore the rectified linear unit (ReLU) is still the most commonly used. This paper proposes a technique for customizing activation functions automatically, resulting in reliable improvements in performance. Evolutionary search is used to discover the general form of the function, and gradient descent to optimize its parameters for different parts of the network …
Institute for Defense Analyses, Center for Communications Research - La Jolla（国防分析研究所拉霍亚通信研究中心）
Neural networks with rectified linear unit activations are essentially multivariate linear splines. As such, one of many ways to measure the "complexity" or "expressivity" of a neural network is to count the number of knots in the spline model. We study the number of knots in fully-connected feedforward neural networks with rectified linear unit activation functions. We intentionally keep the neural networks very simple, so as to make theoretical analyses more approachable. An induction on the number of layers $l$ reveals a tight upper bound on the number of knots in $\mathbb{R} \to \mathbb{R}…
Deep neural networks paved the way for significant improvements in image visual categorization during the last years. However, even though the tasks are highly varying, differing in complexity and difficulty, existing solutions mostly build on the same architectural decisions. This also applies to the selection of activation functions (AFs), where most approaches build on Rectified Linear Units (ReLUs). In this paper, however, we show that the choice of a proper AF has a significant impact on the classification accuracy, in particular, if fine, subtle details are of relevance. Therefore, we pr…
Institute for Cognitive Science（认知科学研究所）；Osnabrück University（奥斯纳布吕克大学）
The most widely used activation functions in current deep feed-forward neural networks are rectified linear units (ReLU), and many alternatives have been successfully applied, as well. However, none of the alternatives have managed to consistently outperform the rest and there is no unified theory connecting properties of the task and network with properties of activation functions for most efficient training. A possible solution is to have the network learn its preferred activation functions. In this work, we introduce Adaptive Blending Units (ABUs), a trainable linear combination of a set of…
An activation function has a significant impact on the efficiency and robustness of the neural networks. As an alternative, we evolved a cutting-edge non-monotonic activation function, Negative Stimulated Hybrid Activation Function (Nish). It acts as a Rectified Linear Unit (ReLU) function for the positive region and a sinus-sigmoidal function for the negative region. In other words, it incorporates a sigmoid and a sine function and gaining new dynamics over classical ReLU. We analyzed the consistency of the Nish for different combinations of essential networks and most common activation funct…
A proper initialization of the weights in a neural network is critical to its convergence. Current insights into weight initialization come primarily from linear activation functions. In this paper, I develop a theory for weight initializations with non-linear activations. First, I derive a general weight initialization strategy for any neural network using activation functions differentiable at 0. Next, I derive the weight initialization strategy for the Rectified Linear Unit (RELU), and provide theoretical insights into why the Xavier initialization is a poor choice with RELU activations. My…
A wide variety of activation functions have been proposed for neural networks. The Rectified Linear Unit (ReLU) is especially popular today. There are many practical reasons that motivate the use of the ReLU. This paper provides new theoretical characterizations that support the use of the ReLU, its variants such as the leaky ReLU, as well as other activation functions in the case of univariate, single-hidden layer feedforward neural networks. Our results also explain the importance of commonly used strategies in the design and training of neural networks such as "weight decay" and "path-norm"…
Rectified Linear Units (ReLUs) are among the most widely used activation function in a broad variety of tasks in vision. Recent theoretical results suggest that despite their excellent practical performance, in various cases, a substitution with basis expansions (e.g., polynomials) can yield significant benefits from both the optimization and generalization perspective. Unfortunately, the existing results remain limited to networks with a couple of layers, and the practical viability of these results is not yet known. Motivated by some of these results, we explore the use of Hermite polynomial…
Consider the following fundamental learning problem: given input examples $x \in \mathbb{R}^d$ and their vector-valued labels, as defined by an underlying generative neural network, recover the weight matrices of this network. We consider two-layer networks, mapping $\mathbb{R}^d$ to $\mathbb{R}^m$, with $k$ non-linear activation units $f(\cdot)$, where $f(x) = \max \{x , 0\}$ is the ReLU. Such a network is specified by two weight matrices, $\mathbf{U}^* \in \mathbb{R}^{m \times k}, \mathbf{V}^* \in \mathbb{R}^{k \times d}$, such that the label of an example $x \in \mathbb{R}^{d}$ is given by …
The choice of activation functions in deep networks has a significant effect on the training dynamics and task performance. Currently, the most successful and widely-used activation function is the Rectified Linear Unit (ReLU). Although various hand-designed alternatives to ReLU have been proposed, none have managed to replace it due to inconsistent gains. In this work, we propose to leverage automatic search techniques to discover new activation functions. Using a combination of exhaustive and reinforcement learning-based search, we discover multiple novel activation functions. We verify the …
Ho Chi Minh City University of Technology, Vietnam National University - Ho Chi Minh（胡志明市理工大学，越南国立大学胡志明市分校）；University of Technology and Education HCMC（胡志明市技术教育大学）；Da Nang University of Science and Technology（岘港科技大学）；Austrian Institute of Technology（奥地利理工学院）
This paper provides an analysis of state-of-the-art activation functions with respect to supervised classification of deep neural network. These activation functions comprise of Rectified Linear Units (ReLU), Exponential Linear Unit (ELU), Scaled Exponential Linear Unit (SELU), Gaussian Error Linear Unit (GELU), and the Inverse Square Root Linear Unit (ISRLU). To evaluate, experiments over two deep learning network architectures integrating these activation functions are conducted. The first model, basing on Multilayer Perceptron (MLP), is evaluated with MNIST dataset to perform these activati…
Gradient-only line searches (GOLS) adaptively determine step sizes along search directions for discontinuous loss functions resulting from dynamic mini-batch sub-sampling in neural network training. Step sizes in GOLS are determined by localizing Stochastic Non-Negative Associated Gradient Projection Points (SNN-GPPs) along descent directions. These are identified by a sign change in the directional derivative from negative to positive along a descent direction. Activation functions are a significant component of neural network architectures as they introduce non-linearities essential for comp…
Sapienza University of Rome（罗马第一大学）；Roma Tre University（罗马第三大学）
The design of activation functions is a growing research area in the field of neural networks. In particular, instead of using fixed point-wise functions (e.g., the rectified linear unit), several authors have proposed ways of learning these functions directly from the data in a non-parametric fashion. In this paper we focus on the kernel activation function (KAF), a recently proposed framework wherein each function is modeled as a one-dimensional kernel model, whose weights are adapted through standard backpropagation-based optimization. One drawback of KAFs is the need to select a single ker…
We propose convex relaxations for convolutional neural nets with one hidden layer where the output weights are fixed. For convex activation functions such as rectified linear units, the relaxations are convex second order cone programs which can be solved very efficiently. We prove that the relaxation recovers the global minimum under a planted model assumption, given sufficiently many training samples from a Gaussian distribution. We also identify a phase transition phenomenon in recovering the global minimum for the relaxation.
Vienna University of Technology（维也纳工业大学）；Politecnico di Torino（都灵理工大学）
Activation functions influence behavior and performance of DNNs. Nonlinear activation functions, like Rectified Linear Units (ReLU), Exponential Linear Units (ELU) and Scaled Exponential Linear Units (SELU), outperform the linear counterparts. However, selecting an appropriate activation function is a challenging problem, as it affects the accuracy and the complexity of the given DNN. In this paper, we propose a novel methodology to automatically select the best-possible activation function for each layer of a given DNN, such that the overall DNN accuracy, compared to considering only one type…
Deep neural networks, as a powerful system to represent high dimensional complex functions, play a key role in deep learning. Convergence of deep neural networks is a fundamental issue in building the mathematical foundation for deep learning. We investigated the convergence of deep ReLU networks and deep convolutional neural networks in two recent researches (arXiv:2107.12530, 2109.13542). Only the Rectified Linear Unit (ReLU) activation was studied therein, and the important pooling strategy was not considered. In this current work, we study the convergence of deep neural networks as the dep…
Mathematical analysis of one-layer neural network with fixed biases, a new activation function and other observations
ASAI, Universidad Politécnica de Madrid. ETSI Navales（马德里理工大学应用智能系统与分析系，海军工程学院）；Department of Mathematics, Faculty of Sciences, Gakushuin University（学习院大学理学院数学系）
Michigan Technological University（密歇根理工大学）；Symbiosis Centre for Medical Image Analysis（共生医学图像分析中心）；Symbiosis Institute of Technology（共生技术学院）
This work introduces a novel activation unit that can be efficiently employed in deep neural nets (DNNs) and performs significantly better than the traditional Rectified Linear Units (ReLU). The function developed is a two parameter version of the specialized Richard's Curve and we call it Adaptive Richard's Curve weighted Activation (ARiA). This function is non-monotonous, analogous to the newly introduced Swish, however allows a precise control over its non-monotonous convexity by varying the hyper-parameters. We first demonstrate the mathematical significance of the two parameter ARiA follo…
In neural networks, non-linearity is introduced by activation functions. One commonly used activation function is Rectified Linear Unit (ReLU). ReLU has been a popular choice as an activation but has flaws. State-of-the-art functions like Swish and Mish are now gaining attention as a better choice as they combat many flaws presented by other activation functions. CoLU is an activation function similar to Swish and Mish in properties. It is defined as f(x)=x/(1-xe^-(x+e^x)). It is smooth, continuously differentiable, unbounded above, bounded below, non-saturating, and non-monotonic. Based on ex…
Texas Southern University（德克萨斯南方大学）；Prairie View A & M University（Prairie View A & M 大学）
The nonlinearity of activation functions used in deep learning models are crucial for the success of predictive models. There are several commonly used simple nonlinear functions, including Rectified Linear Unit (ReLU) and Leaky-ReLU (L-ReLU). In practice, these functions remarkably enhance the model accuracy. However, there is limited insight into the functionality of these nonlinear activation functions in terms of why certain models perform better than others. Here, we investigate the model performance when using ReLU or L-ReLU as activation functions in different model architectures and da…
This paper presents an investigation of the approximation property of neural networks with unbounded activation functions, such as the rectified linear unit (ReLU), which is the new de-facto standard of deep learning. The ReLU network can be analyzed by the ridgelet transform with respect to Lizorkin distributions. By showing three reconstruction formulas by using the Fourier slice theorem, the Radon transform, and Parseval's relation, it is shown that a neural network with unbounded activation functions still satisfies the universal approximation property. As an additional consequence, the ri…
Bernoulli Institute for Mathematics, Computer Science and Artificial Intelligence（伯努利数学、计算机科学与人工智能研究所）；University of Groningen（格罗宁根大学）
We study layered neural networks of rectified linear units (ReLU) in a modelling framework for stochastic training processes. The comparison with sigmoidal activation functions is in the center of interest. We compute typical learning curves for shallow networks with K hidden units in matching student teacher scenarios. The systems exhibit sudden changes of the generalization performance via the process of hidden unit specialization at critical sizes of the training set. Surprisingly, our results show that the training behavior of ReLU networks is qualitatively different from that of networks …
Universiti Tun Hussein Onn Malaysia（马来西亚敦胡先翁大学）；Institute of Engineering and Management（工程与管理学院）
Activation functions are essential for deep learning methods to learn and perform complex tasks such as image classification. Rectified Linear Unit (ReLU) has been widely used and become the default activation function across the deep learning community since 2012. Although ReLU has been popular, however, the hard zero property of the ReLU has heavily hindered the negative values from propagating through the network. Consequently, the deep neural network has not been benefited from the negative representations. In this work, an activation function called Flatten-T Swish (FTS) that leverage the…
We propose ReDense as a simple and low complexity way to improve the performance of trained neural networks. We use a combination of random weights and rectified linear unit (ReLU) activation function to add a ReLU dense (ReDense) layer to the trained neural network such that it can achieve a lower training loss. The lossless flow property (LFP) of ReLU is the key to achieve the lower training loss while keeping the generalization error small. ReDense does not suffer from vanishing gradient problem in the training due to having a shallow structure. We experimentally show that ReDense can impro…
Radboud University（拉德堡德大学）；Donders Institute for Brain, Cognition and Behaviour（东德斯脑、认知与行为研究所）
An important issue in neural network research is how to choose the number of nodes and layers such as to solve a classification problem. We provide new intuitions based on earlier results by An et al. (2015) by deriving an upper bound on the number of nodes in networks with two hidden layers such that linear separability can be achieved. Concretely, we show that if the data can be described in terms of N finite sets and the used activation function f is non-constant, increasing and has a left asymptote, we can derive how many nodes are needed to linearly separate these sets. This will be an up…
Neural networks have been widely used to solve complex real-world problems. Due to the complicate, nonlinear, non-convex nature of neural networks, formal safety guarantees for the output behaviors of neural networks will be crucial for their applications in safety-critical systems.In this paper, the output reachable set computation and safety verification problems for a class of neural networks consisting of Rectified Linear Unit (ReLU) activation functions are addressed. A layer-by-layer approach is developed to compute output reachable set. The computation is formulated in the form of a set…
In a function approximation with a neural network, an input dataset is mapped to an output index by optimizing the parameters of each hidden-layer unit. For a unary function, we present constraints on the parameters and its second derivative by constructing a continuum version of a one-hidden-layer neural network with the rectified linear unit (ReLU) activation function. The network is accurately implemented because the constraints decrease the degrees of freedom of the parameters. We also explain the existence of a smooth binary function that cannot be precisely represented by any such neural…
We consider approximation rates of sparsely connected deep rectified linear unit (ReLU) and rectified power unit (RePU) neural networks for functions in Besov spaces $B^α_{q}(L^p)$ in arbitrary dimension $d$, on general domains. We show that \alert{deep rectifier} networks with a fixed activation function attain optimal or near to optimal approximation rates for functions in the Besov space $B^α_τ(L^τ)$ on the critical embedding line $1/τ=α/d+1/p$ for \emph{arbitrary} smoothness order $α>0$. Using interpolation theory, this implies that the entire range of smoothness classes at or above the cr…
ChebNet: Efficient and Stable Constructions of Deep Neural Networks with Rectified Power Units via Chebyshev Approximations
Software Development Center, Industrial and Commercial Bank of China（中国工商银行软件开发中心）；Hisilicon Semiconductor and Component Business Dept.(2012 Labs), Huawei Technologies Co., Ltd（华为技术有限公司半导体及组件业务部）；NCMIS & LSEC, Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and Systems Science, Beijing（数学与系统科学研究院）；School of Mathematical Sciences, University of Chinese Academy of Sciences（中国科学院大学数学科学学院）
For neural networks (NNs) with rectified linear unit (ReLU) or binary activation functions, we show that their training can be accomplished in a reduced parameter space. Specifically, the weights in each neuron can be trained on the unit sphere, as opposed to the entire space, and the threshold can be trained in a bounded interval, as opposed to the real line. We show that the NNs in the reduced parameter space are mathematically equivalent to the standard NNs with parameters in the whole space. The reduced parameter space shall facilitate the optimization procedure for the network training, a…
