Max-Pooling operations are a core component of deep learning architectures. In particular, they are part of most convolutional architectures used in machine vision, since pooling is a natural approach to pattern detection problems. However, these architectures are not well understood from a theoretical perspective. For example, we do not understand when they can be globally optimized, and what is the effect of over-parameterization on generalization. Here we perform a theoretical analysis of a convolutional max-pooling architecture, proving that it can be globally optimized, and can generalize well even for highly over-parameterized models. Our analysis focuses on a data generating distribution inspired by pattern detection problem, where a "discriminative" pattern needs to be detected among "spurious" patterns. We empirically validate that CNNs significantly outperform fully connected networks in our setting, as predicted by our theoretical results.
Stanford University（斯坦福大学）；University of California, Berkeley（加州大学伯克利分校）
Recent empirical work has shown that hierarchical convolutional kernels inspired by convolutional neural networks (CNNs) significantly improve the performance of kernel methods in image classification tasks. A widely accepted explanation for their success is that these architectures encode hypothesis classes that are suitable for natural images. However, understanding the precise interplay between approximation and generalization in convolutional architectures remains a challenge. In this paper, we consider the stylized setting of covariates (image pixels) uniformly distributed on the hypercub…
Handong Global University（韩东国际大学）；Dept. of Information and Communication Engineering（信息与通信工程系）
Convolutional neural networks (CNNs) have achieved remarkable performance in many applications, especially in image recognition tasks. As a crucial component of CNNs, sub-sampling plays an important role for efficient training or invariance property, and max-pooling and arithmetic average-pooling are commonly used sub-sampling methods. In addition to the two pooling methods, however, there could be many other pooling types, such as geometric average, harmonic average, and so on. Since it is not easy for algorithms to find the best pooling method, usually the pooling types are assumed a priorit…
Image classification is considered, and a hierarchical max-pooling model with additional local pooling is introduced. Here the additional local pooling enables the hierachical model to combine parts of the image which have a variable relative distance towards each other. Various convolutional neural network image classifiers are introduced and compared in view of their rate of convergence. The finite sample size performance of the estimates is analyzed by applying them to simulated and real data.
We propose a max-pooling based loss function for training Long Short-Term Memory (LSTM) networks for small-footprint keyword spotting (KWS), with low CPU, memory, and latency requirements. The max-pooling loss training can be further guided by initializing with a cross-entropy loss trained network. A posterior smoothing based evaluation approach is employed to measure keyword spotting performance. Our experimental results show that LSTM models trained using cross-entropy loss or max-pooling loss outperform a cross-entropy loss trained baseline feed-forward Deep Neural Network (DNN). In additio…
Max- and average-pooling are the most popular pooling methods for downsampling in convolutional neural networks. In this paper, we compare different pooling methods that generalize both max- and average-pooling. Furthermore, we propose another method based on a smooth approximation of the maximum function and put it into context with related methods. For the comparison, we use a VGG16 image classification network and train it on a large dataset of natural high-resolution images (Google Open Images v5). The results show that none of the more sophisticated methods perform significantly better in…
Recently, dropout has seen increasing use in deep learning. For deep convolutional neural networks, dropout is known to work well in fully-connected layers. However, its effect in pooling layers is still not clear. This paper demonstrates that max-pooling dropout is equivalent to randomly picking activation based on a multinomial distribution at training time. In light of this insight, we advocate employing our proposed probabilistic weighted pooling, instead of commonly used max-pooling, to act as model averaging at test time. Empirical evidence validates the superiority of probabilistic weig…
Harbin Institute of Technology（哈尔滨工业大学）；Peking University（北京大学）
In most convolution neural networks (CNNs), downsampling hidden layers is adopted for increasing computation efficiency and the receptive field size. Such operation is commonly so-called pooling. Maximation and averaging over sliding windows (max/average pooling), and plain downsampling in the form of strided convolution are popular pooling methods. Since the pooling is a lossy procedure, a motivation of our work is to design a new pooling approach for less lossy in the dimensionality reduction. Inspired by the Fourier spectral pooling(FSP) proposed by Rippel et. al. [1], we present the Hartle…
MaxCutPool: differentiable feature-aware Maxcut for pooling in graph neural networks
Alma Mater Studiorum - University of Bologna（博洛尼亚大学）；UiT the Arctic University of Norway（挪威北极大学）；Fondazione Istituto Italiano di Tecnologia（意大利技术研究院）；NORCE Norwegian Research Centre AS（NORCE挪威研究中心）
Idiap Research Institute（Idiap研究所）；EPFL（洛桑联邦理工学院）；University of Geneva（日内瓦大学）
Over the decade since deep neural networks became state of the art image classifiers there has been a tendency towards less use of max pooling: the function that takes the largest of nearby pixels in an image. Since max pooling featured prominently in earlier generations of image classifiers, we wish to understand this trend, and whether it is justified. We develop a theoretical framework analyzing ReLU based approximations to max pooling, and prove a sense in which max pooling cannot be efficiently replicated using ReLU activations. We analyze the error of a class of optimal approximations, a…
The School of Computer Science（计算机科学学院）；Tel Aviv University（特拉维夫大学）；Facebook AI Research（脸书人工智能研究院）
While there are optimal TSP solvers, as well as recent learning-based approaches, the generalization of the TSP to the Multiple Traveling Salesmen Problem is much less studied. Here, we design a neural network solution that treats the salesmen, cities and depot as three different sets of varying cardinalities. We apply a novel technique that combines elements from recent architectures that were developed for sets, as well as elements from graph networks. Coupled with new constraint enforcing output layers, a dedicated loss, and a search method, our solution is shown to outperform all the meta-…
We seek to improve deep neural networks by generalizing the pooling operations that play a central role in current architectures. We pursue a careful exploration of approaches to allow pooling to learn and to adapt to complex and variable patterns. The two primary directions lie in (1) learning a pooling function via (two strategies of) combining of max and average pooling, and (2) learning a pooling function in the form of a tree-structured fusion of pooling filters that are themselves learned. In our experiments every generalized pooling operation we explore improves performance when used in…
Recently many effective attention modules are proposed to boot the model performance by exploiting the internal information of convolutional neural networks in computer vision. In general, many previous works ignore considering the design of the pooling strategy of the attention mechanism since they adopt the global average pooling for granted, which hinders the further improvement of the performance of the attention mechanism. However, we empirically find and verify a phenomenon that the simple linear combination of global max-pooling and global min-pooling can produce pooling strategies that…
Indian Institute of Technology, Delhi（德里印度理工学院）；Carnegie Mellon University（卡内基梅隆大学）
Pooling-based recurrent neural architectures consistently outperform their counterparts without pooling. However, the reasons for their enhanced performance are largely unexamined. In this work, we examine three commonly used pooling techniques (mean-pooling, max-pooling, and attention), and propose max-attention, a novel variant that effectively captures interactions among predictive tokens in a sentence. We find that pooling-based architectures substantially differ from their non-pooling equivalents in their learning ability and positional biases--which elucidate their performance benefits. …
Enumeration of max-pooling responses with generalized permutohedra
Washington University in St. Louis（圣路易斯华盛顿大学）；University of California, Riverside（加州大学河滨分校）；Harvey Mudd College（哈维穆德学院）；University of California, Los Angeles（加州大学洛杉矶分校）；Max Planck Institute for Mathematics in the Sciences（马克斯·普朗克数学科学研究所）；Université du Québec à Montréal（魁北克大学蒙特利尔分校）
We present in this paper a simple, yet efficient convolutional neural network (CNN) architecture for robust audio event recognition. Opposing to deep CNN architectures with multiple convolutional and pooling layers topped up with multiple fully connected layers, the proposed network consists of only three layers: convolutional, pooling, and softmax layer. Two further features distinguish it from the deep architectures that have been proposed for the task: varying-size convolutional filters at the convolutional layer and 1-max pooling scheme at the pooling layer. In intuition, the network tends…
In modern computer vision tasks, convolutional neural networks (CNNs) are indispensable for image classification tasks due to their efficiency and effectiveness. Part of their superiority compared to other architectures, comes from the fact that a single, local filter is shared across the entire image. However, there are scenarios where we may need to treat spatial locations in non-uniform manner. We see this in nature when considering how humans have evolved foveation to process different areas in their field of vision with varying levels of detail. In this paper we propose a way to enable CN…
We present generalized versions of the commonly used maximum pooling operation: $k$th maximum and sorted pooling operations which selects the $k$th largest response in each pooling region, selecting locally consistent features of the input images. This method is able to increase the generalization power of a network and can be used to decrease training time and error rate of networks and it can significantly improve accuracy in case of training scenarios where the amount of available data is limited, like one-shot learning scenarios
Many theories have emerged which investigate how in- variance is generated in hierarchical networks through sim- ple schemes such as max and mean pooling. The restriction to max/mean pooling in theoretical and empirical studies has diverted attention away from a more general way of generating invariance to nuisance transformations. We con- jecture that hierarchically building selective invariance (i.e. carefully choosing the range of the transformation to be in- variant to at each layer of a hierarchical network) is im- portant for pattern recognition. We utilize a novel pooling layer called a…
In this paper, we propose a novel pooling layer for graph neural networks based on maximizing the mutual information between the pooled graph and the input graph. Since the maximum mutual information is difficult to compute, we employ the Shannon capacity of a graph as an inductive bias to our pooling method. More precisely, we show that the input graph to the pooling layer can be viewed as a representation of a noisy communication channel. For such a channel, sending the symbols belonging to an independent set of the graph yields a reliable and error-free transmission of information. We show …
In convolutional neural networks (CNNs), pooling operations play important roles such as dimensionality reduction and deformation compensation. In general, max pooling, which is the most widely used operation for local pooling, is performed independently for each kernel. However, the deformation may be spatially smooth over the neighboring kernels. This means that max pooling is too flexible to compensate for actual deformations. In other words, its excessive flexibility risks canceling the essential spatial differences between classes. In this paper, we propose regularized pooling, which enab…
Xinjiang University（新疆大学）；Software College, Xinjiang University（新疆大学软件学院）；Network Center, Xinjiang University（新疆大学网络中心）
To better retain the deep features of an image and solve the sparsity problem of the end-to-end segmentation model, we propose a new deep convolutional network model for medical image pixel segmentation, called MC-Net. The core of this network model consists of four parts, namely, an encoder network, a multiple max-pooling integration module, a cross multiscale deconvolution decoder network and a pixel-level classification layer. In the network structure of the encoder, we use multiscale convolution instead of the traditional single-channel convolution. The multiple max-pooling integration mod…
Recently, dropout has seen increasing use in deep learning. For deep convolutional neural networks, dropout is known to work well in fully-connected layers. However, its effect in convolutional and pooling layers is still not clear. This paper demonstrates that max-pooling dropout is equivalent to randomly picking activation based on a multinomial distribution at training time. In light of this insight, we advocate employing our proposed probabilistic weighted pooling, instead of commonly used max-pooling, to act as model averaging at test time. Empirical evidence validates the superiority of …
This work generalizes graph neural networks (GNNs) beyond those based on the Weisfeiler-Lehman (WL) algorithm, graph Laplacians, and diffusions. Our approach, denoted Relational Pooling (RP), draws from the theory of finite partial exchangeability to provide a framework with maximal representation power for graphs. RP can work with existing graph representation models and, somewhat counterintuitively, can make them even more powerful than the original WL isomorphism test. Additionally, RP allows architectures like Recurrent Neural Networks and Convolutional Neural Networks to be used in a theo…
Physical Pooling Functions in Graph Neural Networks for Molecular Property Prediction
RWTH Aachen University（亚琛工业大学）；Delft University of Technology（代尔夫特理工大学）；Forschungszentrum Jülich GmbH（于利希研究中心）；JARA Center for Simulation and Data Science (CSD)（JARA模拟与数据科学中心）
From Correlation to Causation: Max-Pooling-Based Multi-Instance Learning Leads to More Robust Whole Slide Image Classification
UniSA STEM, University of South Australia（南澳大利亚大学STEM学院）；School of Computer Science and Engineering, Southeast University（东南大学计算机科学与工程学院）；Key Lab. of Computer Network and Information Integration (Southeast University), MoE, China（计算机网络与信息集成重点实验室（东南大学），教育部，中国）；School of Information and Physical Sciences, The University of Newcastle（新castle大学信息与物理科学学院）；Mohamed bin Zayed University of Artificial Intelligence (MBZUAI)（穆罕默德·本·扎耶德人工智能大学）
A longstanding goal in the theory of deep learning is to characterize the conditions under which a given neural network architecture will be trainable, and if so, how well it might generalize to unseen data. In this work, we provide such a characterization in the limit of very wide and very deep networks, for which the analysis simplifies considerably. For wide networks, the trajectory under gradient descent is governed by the Neural Tangent Kernel (NTK), and for deep networks the NTK itself maintains only weak data dependence. By analyzing the spectrum of the NTK, we formulate necessary condi…
We propose a novel pooling strategy that learns how to adaptively rank deep convolutional features for selecting more informative representations. To this end, we exploit discriminative analysis to project the features onto a space spanned by the number of classes in the dataset under study. This maps the notion of labels in the feature space into instances in the projected space. We employ these projected distances as a measure to rank the existing features with respect to their specific discriminant power for each individual class. We then apply multipartite ranking to score the separability…
The empirical success of deep convolutional networks on tasks involving high-dimensional data such as images or audio suggests that they can efficiently approximate certain functions that are well-suited for such tasks. In this paper, we study this through the lens of kernel methods, by considering simple hierarchical kernels with two or three convolution and pooling layers, inspired by convolutional kernel networks. These achieve good empirical performance on standard vision datasets, while providing a precise description of their functional space that yields new insights on their inductive b…
TU Darmstadt（达姆施塔特工业大学）；NEC Laboratories Europe（欧洲NEC实验室）；Oculus Research（Oculus研究院）
Most convolutional neural networks use some method for gradually downscaling the size of the hidden layers. This is commonly referred to as pooling, and is applied to reduce the number of parameters, improve invariance to certain distortions, and increase the receptive field size. Since pooling by nature is a lossy process, it is crucial that each such layer maintains the portion of the activations that is most important for the network's discriminability. Yet, simple maximization or averaging over blocks, max or average pooling, or plain downsampling in the form of strided convolutions are th…
Faculty of Mathematics and Computer Science Jagiellonian University（雅盖隆大学数学与计算机科学学院）；Ardigen
Global pooling, such as max- or sum-pooling, is one of the key ingredients in deep neural networks used for processing images, texts, graphs and other types of structured data. Based on the recent DeepSets architecture proposed by Zaheer et al. (NIPS 2017), we introduce a Set Aggregation Network (SAN) as an alternative global pooling layer. In contrast to typical pooling operators, SAN allows to embed a given set of features to a vector representation of arbitrary size. We show that by adjusting the size of embedding, SAN is capable of preserving the whole information from the input. In experi…
Pooling is essentially an operation from the field of Mathematical Morphology, with max pooling as a limited special case. The more general setting of MorphPooling greatly extends the tool set for building neural networks. In addition to pooling operations, encoder-decoder networks used for pixel-level predictions also require unpooling. It is common to combine unpooling with convolution or deconvolution for up-sampling. However, using its morphological properties, unpooling can be generalised and improved. Extensive experimentation on two tasks and three large-scale datasets shows that morpho…
Graph neural networks (GNN) extends deep learning to graph-structure dataset. Similar to Convolutional Neural Networks (CNN) using on image prediction, convolutional and pooling layers are the foundation to success for GNN on graph prediction tasks. In the initial PAN paper, it uses a path integral based graph neural networks for graph prediction. Specifically, it uses a convolution operation that involves every path linking the message sender and receiver with learnable weights depending on the path length, which corresponds to the maximal entropy random walk. It further generalizes such conv…
The goal of this paper is to introduce pooling strategies for simplicial convolutional neural networks. Inspired by graph pooling methods, we introduce a general formulation for a simplicial pooling layer that performs: i) local aggregation of simplicial signals; ii) principled selection of sampling sets; iii) downsampling and simplicial topology adaptation. The general layer is then customized to design four different pooling strategies (i.e., max, top-k, self-attention, and separated top-k) grounded in the theory of topological signal processing. Also, we leverage the proposed layers in a hi…
Colorado State University（科罗拉多州立大学）；The Ohio State University（俄亥俄州立大学）；Boise State University（博伊西州立大学）；Cooperative Institute for Research in the Atmosphere（大气研究合作研究所）
Machine learning-based analysis of medical images often faces several hurdles, such as the lack of training data, the curse of dimensionality problem, and the generalization issues. One of the main difficulties is that there exists computational cost problem in dealing with input data of large size matrices which represent medical images. The purpose of this paper is to introduce a framelet-pooling aided deep learning method for mitigating computational bundle, caused by large dimensionality. By transforming high dimensional data into low dimensional components by filter banks with preserving …
National Yang Ming Chiao Tung University（阳明交通大学）；University at Albany - SUNY（奥尔巴尼大学 - 纽约州立大学）；University at Buffalo（布法罗大学）
Pooling is a simple but essential layer in modern deep CNN architectures for feature aggregation and extraction. Typical CNN design focuses on the conv layers and activation functions, while leaving the pooling layers with fewer options. We introduce the Learning Discrete Wavelet Pooling (LDW-Pooling) that can be applied universally to replace standard pooling operations to better extract features with improved accuracy and efficiency. Motivated from the wavelet theory, we adopt the low-pass (L) and high-pass (H) filters horizontally and vertically for pooling on a 2D feature map. Feature sign…
Institute of Computing Technology, CAS（中国科学院计算技术研究所）；University of Chinese Academy of Sciences（中国科学院大学）；Cardiff University（卡迪夫大学）；City University of Hong Kong（香港城市大学）
3D shape analysis is an important research topic in computer vision and graphics. While existing methods have generalized image-based deep learning to meshes using graph-based convolutions, the lack of an effective pooling operation restricts the learning capability of their networks. In this paper, we propose a novel pooling operation for mesh datasets with the same connectivity but different geometry, by building a mesh hierarchy using mesh simplification. For this purpose, we develop a modified mesh simplification method to avoid generating highly irregularly sized triangles. Our pooling op…
Pooling operations, which can be calculated at low cost and serve as a linear or nonlinear transfer function for data reduction, are found in almost every modern neural network. Countless modern approaches have already tackled replacing the common maximum value selection and mean value operations, not to mention providing a function that allows different functions to be selected through changing parameters. Additional neural networks are used to estimate the parameters of these pooling functions.Consequently, pooling layers may require supplementary parameters to increase the complexity of the…
Dalian University of Technology（大连理工大学）；Tianjin University（天津大学）
For video recognition task, a global representation summarizing the whole contents of the video snippets plays an important role for the final performance. However, existing video architectures usually generate it by using a simple, global average pooling (GAP) method, which has limited ability to capture complex dynamics of videos. For image recognition task, there exist evidences showing that covariance pooling has stronger representation ability than GAP. Unfortunately, such plain covariance pooling used in image recognition is an orderless representative, which cannot model spatio-temporal…
In this paper, we deal with two challenges for measuring the similarity of the subject identities in practical video-based face recognition - the variation of the head pose in uncontrolled environments and the computational expense of processing videos. Since the frame-wise feature mean is unable to characterize the pose diversity among frames, we define and preserve the overall pose diversity and closeness in a video. Then, identity will be the only source of variation across videos since the pose varies even within a single video. Instead of simply using all the frames, we select those faces…
Dalhousie University（达尔豪斯大学）；Vector Institute for Artificial Intelligence（向量人工智能研究所）
We seek to improve the pooling operation in neural networks, by applying a more theoretically justified operator. We demonstrate that LogSumExp provides a natural OR operator for logits. When one corrects for the number of elements inside the pooling operator, this becomes $\text{LogAvgExp} := \log(\text{mean}(\exp(x)))$. By introducing a single temperature parameter, LogAvgExp smoothly transitions from the max of its operands to the mean (found at the limiting cases $t \to 0^+$ and $t \to +\infty$). We experimentally tested LogAvgExp, both with and without a learnable temperature parameter, i…
In this paper we propose and investigate a novel nonlinear unit, called $L_p$ unit, for deep neural networks. The proposed $L_p$ unit receives signals from several projections of a subset of units in the layer below and computes a normalized $L_p$ norm. We notice two interesting interpretations of the $L_p$ unit. First, the proposed unit can be understood as a generalization of a number of conventional pooling operators such as average, root-mean-square and max pooling widely used in, for instance, convolutional neural networks (CNN), HMAX models and neocognitrons. Furthermore, the $L_p$ unit …
This paper reports a novel deep architecture referred to as Maxout network In Network (MIN), which can enhance model discriminability and facilitate the process of information abstraction within the receptive field. The proposed network adopts the framework of the recently developed Network In Network structure, which slides a universal approximator, multilayer perceptron (MLP) with rectifier units, to exact features. Instead of MLP, we employ maxout MLP to learn a variety of piecewise linear activation functions and to mediate the problem of vanishing gradients that can occur when using recti…
Princeton University（普林斯顿大学）；University of Technology Sydney（悉尼科技大学）；Max Planck Institute for Mathematics in the Sciences（马克斯·普朗克数学科学研究所）；University of New South Wales（新南威尔士大学）；Zhejiang Normal University（浙江师范大学）；University of Cambridge（剑桥大学）
Graph neural networks (GNNs) extends the functionality of traditional neural networks to graph-structured data. Similar to CNNs, an optimized design of graph convolution and pooling is key to success. Borrowing ideas from physics, we propose a path integral based graph neural networks (PAN) for classification and regression tasks on graphs. Specifically, we consider a convolution operation that involves every path linking the message sender and receiver with learnable weights depending on the path length, which corresponds to the maximal entropy random walk. It generalizes the graph Laplacian …
Music and Audio Research Laboratory（音乐与音频研究实验室）；Center for Data Science（数据科学中心）；Center for Urban Science and Progress（城市科学与进步中心）；New York University（纽约大学）
Sound event detection (SED) methods are tasked with labeling segments of audio recordings by the presence of active sound sources. SED is typically posed as a supervised machine learning problem, requiring strong annotations for the presence or absence of each sound source at every time instant within the recording. However, strong annotations of this type are both labor- and cost-intensive for human annotators to produce, which limits the practical scalability of SED methods. In this work, we treat SED as a multiple instance learning (MIL) problem, where training labels are static over a shor…
Beihang University（北京航空航天大学）；Macquarie University（麦考瑞大学）；University of Illinois at Chicago（伊利诺伊大学芝加哥分校）；Lehigh University（理海大学）
Graph representation learning has attracted increasing research attention. However, most existing studies fuse all structural features and node attributes to provide an overarching view of graphs, neglecting finer substructures' semantics, and suffering from interpretation enigmas. This paper presents a novel hierarchical subgraph-level selection and embedding based graph neural network for graph classification, namely SUGAR, to learn more discriminative subgraph representations and respond in an explanatory way. SUGAR reconstructs a sketched graph by extracting striking subgraphs as the repre…
McGill University（麦吉尔大学）；Université de Montréal（蒙特利尔大学）；HEC Montréal（蒙特利尔高等商学院）；Mila（米拉人工智能研究所）
Graph Neural Networks (GNNs) are attracting growing attention due to their effectiveness and flexibility in modeling a variety of graph-structured data. Exiting GNN architectures usually adopt simple pooling operations (eg. sum, average, max) when aggregating messages from a local neighborhood for updating node representation or pooling node representations from the entire graph to compute the graph representation. Though simple and effective, these linear operations do not model high-order non-linear interactions among nodes. We propose the Tensorized Graph Neural Network (tGNN), a highly exp…
Binghamton University（宾汉姆顿大学）；IBM T.J. Watson Research Center（IBM T.J. 沃森研究中心）；University of California, San Diego（加州大学圣地亚哥分校）
Feature pooling layers (e.g., max pooling) in convolutional neural networks (CNNs) serve the dual purpose of providing increasingly abstract representations as well as yielding computational savings in subsequent convolutional layers. We view the pooling operation in CNNs as a two-step procedure: first, a pooling window (e.g., $2\times 2$) slides over the feature map with stride one which leaves the spatial resolution intact, and second, downsampling is performed by selecting one pixel from each non-overlapping pooling window in an often uniform and deterministic (e.g., top-left) manner. Our s…
Ant Group（蚂蚁集团）；Wuhan University（武汉大学）；Swinburne University of Technology（斯威本科技大学）
Heterogeneous graph neural network has unleashed great potential on graph representation learning and shown superior performance on downstream tasks such as node classification and clustering. Existing heterogeneous graph learning networks are primarily designed to either rely on pre-defined meta-paths or use attention mechanisms for type-specific attentive message propagation on different nodes/edges, incurring many customization efforts and computational costs. To this end, we design a relation-centered Pooling and Convolution for Heterogeneous Graph learning Network, namely PC-HGN, to enabl…
MINES Paris（巴黎矿业学院）；PSL University（巴黎文理研究大学）；Centre for mathematical morphology (CMM)（数学形态学中心（CMM））
This paper analyses both nonlinear activation functions and spatial max-pooling for Deep Convolutional Neural Networks (DCNNs) by means of the algebraic basis of mathematical morphology. Additionally, a general family of activation functions is proposed by considering both max-pooling and nonlinear operators in the context of morphological representations. Experimental section validates the goodness of our approach on classical benchmarks for supervised learning by DCNN.
