We propose reproducing activation functions (RAFs) to improve deep learning accuracy for various applications ranging from computer vision to scientific computing. The idea is to employ several basic functions and their learnable linear combination to construct neuron-wise data-driven activation functions for each neuron. Armed with RAFs, neural networks (NNs) can reproduce traditional approximation tools and, therefore, approximate target functions with a smaller number of parameters than traditional NNs. In NN training, RAFs can generate neural tangent kernels (NTKs) with a better condition number than traditional activation functions lessening the spectral bias of deep learning. As demonstrated by extensive numerical tests, the proposed RAFs can facilitate the convergence of deep learning optimization for a solution with higher accuracy than existing deep learning solvers for audio/image/video reconstruction, PDEs, and eigenvalue problems. With RAFs, the errors of audio/video reconstruction, PDEs, and eigenvalue problems are decreased by over 14%, 73%, 99%, respectively, compared with baseline, while the performance of image reconstruction increases by 58%.
ETH Zürich（苏黎世联邦理工学院）；Università di Genova（热那亚大学）；MIT（麻省理工学院）；IIT（意大利理工学院）
Characterizing the function spaces corresponding to neural networks can provide a way to understand their properties. In this paper we discuss how the theory of reproducing kernel Banach spaces can be used to tackle this challenge. In particular, we prove a representer theorem for a wide class of reproducing kernel Banach spaces that admit a suitable integral representation and include one hidden layer neural networks of possibly infinite width. Further, we show that, for a suitable class of ReLU activation functions, the norm in the corresponding reproducing kernel Banach space can be charact…
Deep learning researchers have a keen interest in proposing two new novel activation functions which can boost network performance. A good choice of activation function can have significant consequences in improving network performance. A handcrafted activation is the most common choice in neural network models. ReLU is the most common choice in the deep learning community due to its simplicity though ReLU has some serious drawbacks. In this paper, we have proposed a new novel activation function based on approximation of known activation functions like Leaky ReLU, and we call this function Sm…
We have proposed orthogonal-Padé activation functions, which are trainable activation functions and show that they have faster learning capability and improves the accuracy in standard deep learning datasets and models. Based on our experiments, we have found two best candidates out of six orthogonal-Padé activations, which we call safe Hermite-Pade (HP) activation functions, namely HP-1 and HP-2. When compared to ReLU, HP-1 and HP-2 has an increment in top-1 accuracy by 5.06% and 4.63% respectively in PreActResNet-34, by 3.02% and 2.75% respectively in MobileNet V2 model on CIFAR100 dataset w…
Neural reproducing kernel Banach spaces and representer theorems for deep networks
Analysis Group - Delft Institute of Applied Mathematics, TU Delft, Netherlands；MaLGa - DIMA, University of Genoa, Italy；MaLGa - DIBRIS, University of Genoa, Italy \& CBMM, MIT \& IIT；RoMaDS - Department of Mathematics, University of Rome Tor Vergata, Italy
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…
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…
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…
Deep Asymmetric Networks with a Set of Node-wise Variant Activation Functions
Electronics and Telecommunications Research Institute (ETRI)（电子与电信研究所（ETRI））；Ulsan National Institute of Science and Technology (UNIST)（蔚山科学技术院（UNIST））
University of Oxford（牛津大学）；The Alan Turing Institute（阿兰·图灵研究所）
Deep Convolutional Sparse Coding (D-CSC) is a framework reminiscent of deep convolutional neural networks (DCNNs), but by omitting the learning of the dictionaries one can more transparently analyse the role of the activation function and its ability to recover activation paths through the layers. Papyan, Romano, and Elad conducted an analysis of such an architecture, demonstrated the relationship with DCNNs and proved conditions under which the D-CSC is guaranteed to recover specific activation paths. A technical innovation of their work highlights that one can view the efficacy of the ReLU n…
Artificial neural networks (ANN), typically referred to as neural networks, are a class of Machine Learning algorithms and have achieved widespread success, having been inspired by the biological structure of the human brain. Neural networks are inherently powerful due to their ability to learn complex function approximations from data. This generalization ability has been able to impact multidisciplinary areas involving image recognition, speech recognition, natural language processing, and others. Activation functions are a crucial sub-component of neural networks. They define the output of …
Samsung India Research Bangalore（三星印度班加罗尔研究所）；International Institute of Information Technology（国际信息技术学院）
Success of deep neural networks in diverse tasks across domains of computer vision, speech recognition and natural language processing, has necessitated understanding the dynamics of training process and also working of trained models. Two independent contributions of this paper are 1) Novel activation function for faster training convergence 2) Systematic pruning of filters of models trained irrespective of activation function. We analyze the topological transformation of the space of training samples as it gets transformed by each successive layer during training, by changing the activation …
RWTH Aachen University（亚琛工业大学）；African Institute for Mathematical Sciences（非洲数学科学研究所）；Stellenbosch University（斯泰伦博斯大学）
We study deep neural networks with binary activation functions (BDNN), i.e. the activation function only has two states. We show that the BDNN can be reformulated as a mixed-integer linear program which can be solved to global optimality by classical integer programming solvers. Additionally, a heuristic solution algorithm is presented and we study the model under data uncertainty, applying a two-stage robust optimization approach. We implemented our methods on random and real datasets and show that the heuristic version of the BDNN outperforms classical deep neural networks on the Breast Canc…
University of Toronto（多伦多大学）；Stony Brook University（纽约州立大学石溪分校）
Activation functions play a pivotal role in determining the training dynamics and neural network performance. The widely adopted activation function ReLU despite being simple and effective has few disadvantages including the Dying ReLU problem. In order to tackle such problems, we propose a novel activation function called Serf which is self-regularized and nonmonotonic in nature. Like Mish, Serf also belongs to the Swish family of functions. Based on several experiments on computer vision (image classification and object detection) and natural language processing (machine translation, sentime…
College of Computer Science, Faculty of Information Technology（信息技术学院计算机学院）；Beijing University of Technology（北京工业大学）
In this paper, it is shown theoretically that spurious local minima are common for deep fully-connected networks and convolutional neural networks (CNNs) with piecewise linear activation functions and datasets that cannot be fitted by linear models. A motivating example is given to explain the reason for the existence of spurious local minima: each output neuron of deep fully-connected networks and CNNs with piecewise linear activations produces a continuous piecewise linear (CPWL) output, and different pieces of CPWL output can fit disjoint groups of data samples when minimizing the empirical…
Mathematical Institute, University of Oxford（牛津大学数学研究所）；The Alan Turing Institute（阿兰·图灵研究所）
The activation function deployed in a deep neural network has great influence on the performance of the network at initialisation, which in turn has implications for training. In this paper we study how to avoid two problems at initialisation identified in prior works: rapid convergence of pairwise input correlations, and vanishing and exploding gradients. We prove that both these problems can be avoided by choosing an activation function possessing a sufficiently large linear region around the origin, relative to the bias variance $σ_b^2$ of the network's random initialisation. We demonstrate…
Privacy concerns in client-server machine learning have given rise to private inference (PI), where neural inference occurs directly on encrypted inputs. PI protects clients' personal data and the server's intellectual property. A common practice in PI is to use garbled circuits to compute nonlinear functions privately, namely ReLUs. However, garbled circuits suffer from high storage, bandwidth, and latency costs. To mitigate these issues, PI-friendly polynomial activation functions have been employed to replace ReLU. In this work, we ask: Is it feasible to substitute all ReLUs with low-degree…
We utilize a connection between compositional kernels and branching processes via Mehler's formula to study deep neural networks. This new probabilistic insight provides us a novel perspective on the mathematical role of activation functions in compositional neural networks. We study the unscaled and rescaled limits of the compositional kernels and explore the different phases of the limiting behavior, as the compositional depth increases. We investigate the memorization capacity of the compositional kernels and neural networks by characterizing the interplay among compositional depth, sample …
We propose a new type of neural networks, Kronecker neural networks (KNNs), that form a general framework for neural networks with adaptive activation functions. KNNs employ the Kronecker product, which provides an efficient way of constructing a very wide network while keeping the number of parameters low. Our theoretical analysis reveals that under suitable conditions, KNNs induce a faster decay of the loss than that by the feed-forward networks. This is also empirically verified through a set of computational examples. Furthermore, under certain technical assumptions, we establish global co…
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…
Old Dominion University（老道明大学）；Sun Yat-sen University（中山大学）
We explore convergence of deep neural networks with the popular ReLU activation function, as the depth of the networks tends to infinity. To this end, we introduce the notion of activation domains and activation matrices of a ReLU network. By replacing applications of the ReLU activation function by multiplications with activation matrices on activation domains, we obtain an explicit expression of the ReLU network. We then identify the convergence of the ReLU networks as convergence of a class of infinite products of matrices. Sufficient and necessary conditions for convergence of these infini…
Deep Learning-Based Survival Analysis with Copula-Based Activation Functions for Multivariate Response Prediction
Statistics Discipline, Division of Science and Mathematics, University of Minnesota-Morris（明尼苏达州立大学莫里斯分校统计学系）；EGADE Business School, Tecnológico de Monterrey（墨西哥蒙特雷技术学院EGADE商学院）；Department of Statistics and Data Science, Pukyong National University（浦项国立大学统计学与数据科学系）；Department of Management Information Systems, Dong-A University（东阿大学管理信息系）
Activation functions shape the outputs of artificial neurons and, therefore, are integral parts of neural networks in general and deep learning in particular. Some activation functions, such as logistic and relu, have been used for many decades. But with deep learning becoming a mainstream research topic, new activation functions have mushroomed, leading to confusion in both theory and practice. This paper provides an analytic yet up-to-date overview of popular activation functions and their properties, which makes it a timely resource for anyone who studies or applies neural networks.
