Global pooling is one of the most significant operations in many machine learning models and tasks, which works for information fusion and structured data (like sets and graphs) representation. However, without solid mathematical fundamentals, its practical implementations often depend on empirical mechanisms and thus lead to sub-optimal, even unsatisfactory performance. In this work, we develop a novel and generalized global pooling framework through the lens of optimal transport. The proposed framework is interpretable from the perspective of expectation-maximization. Essentially, it aims at learning an optimal transport across sample indices and feature dimensions, making the corresponding pooling operation maximize the conditional expectation of input data. We demonstrate that most existing pooling methods are equivalent to solving a regularized optimal transport (ROT) problem with different specializations, and more sophisticated pooling operations can be implemented by hierarchically solving multiple ROT problems. Making the parameters of the ROT problem learnable, we develop a family of regularized optimal transport pooling (ROTP) layers. We implement the ROTP layers as a new kind of deep implicit layer. Their model architectures correspond to different optimization algorithms. We test our ROTP layers in several representative set-level machine learning scenarios, including multi-instance learning (MIL), graph classification, graph set representation, and image classification. Experimental results show that applying our ROTP layers can reduce the difficulty of the design and selection of global pooling -- our ROTP layers may either imitate some existing global pooling methods or lead to some new pooling layers fitting data better. The code is available at \url{https://github.com/SDS-Lab/ROT-Pooling}.
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…
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…
Yale University（耶鲁大学）；Yale School of Medicine（耶鲁大学医学院）；Facebook AI Research（Facebook人工智能研究院）
Understanding how certain brain regions relate to a specific neurological disorder has been an important area of neuroimaging research. A promising approach to identify the salient regions is using Graph Neural Networks (GNNs), which can be used to analyze graph structured data, e.g. brain networks constructed by functional magnetic resonance imaging (fMRI). We propose an interpretable GNN framework with a novel salient region selection mechanism to determine neurological brain biomarkers associated with disorders. Specifically, we design novel regularized pooling layers that highlight salient…
Gaoling School of Artificial Intelligence（高瓴人工智能学院）；Renmin University of China（中国人民大学）
Global pooling is one of the most significant operations in many machine learning models and tasks, whose implementation, however, is often empirical in practice. In this study, we develop a novel and solid global pooling framework through the lens of optimal transport. We demonstrate that most existing global pooling methods are equivalent to solving some specializations of an unbalanced optimal transport (UOT) problem. Making the parameters of the UOT problem learnable, we unify various global pooling methods in the same framework, and accordingly, propose a generalized global pooling layer …
We study training of Convolutional Neural Networks (CNNs) with ReLU activations and introduce exact convex optimization formulations with a polynomial complexity with respect to the number of data samples, the number of neurons, and data dimension. More specifically, we develop a convex analytic framework utilizing semi-infinite duality to obtain equivalent convex optimization problems for several two- and three-layer CNN architectures. We first prove that two-layer CNNs can be globally optimized via an $\ell_2$ norm regularized convex program. We then show that multi-layer circular CNN traini…
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…
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
We propose a novel graph pooling operation using cliques as the unit pool. As this approach is purely topological, rather than featural, it is more readily interpretable, a better analogue to image coarsening than filtering or pruning techniques, and entirely nonparametric. The operation is implemented within graph convolution network (GCN) and GraphSAGE architectures and tested against standard graph classification benchmarks. In addition, we explore the backwards compatibility of the pooling to regular graphs, demonstrating competitive performance when replacing two-by-two pooling in standar…
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…
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…
Graph Neural Networks (GNNs), which generalize deep neural networks to graph-structured data, have drawn considerable attention and achieved state-of-the-art performance in numerous graph related tasks. However, existing GNN models mainly focus on designing graph convolution operations. The graph pooling (or downsampling) operations, that play an important role in learning hierarchical representations, are usually overlooked. In this paper, we propose a novel graph pooling operator, called Hierarchical Graph Pooling with Structure Learning (HGP-SL), which can be integrated into various graph n…
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…
The size and shape of the receptive field determine how the network aggregates local information and affect the overall performance of a model considerably. Many components in a neural network, such as kernel sizes and strides for convolution and pooling operations, influence the configuration of a receptive field. However, they still rely on hyperparameters, and the receptive fields of existing models result in suboptimal shapes and sizes. Hence, we propose a simple yet effective Dynamically Optimized Pooling operation, referred to as DynOPool, which optimizes the scale factors of feature map…
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…
Graph neural networks (GNNs) have been used effectively in different applications involving the processing of signals on irregular structures modeled by graphs. Relying on the use of shift-invariant graph filters, GNNs extend the operation of convolution to graphs. However, the operations of pooling and sampling are still not clearly defined and the approaches proposed in the literature either modify the graph structure in a way that does not preserve its spectral properties, or require defining a policy for selecting which nodes to keep. In this work, we propose a new strategy for pooling and…
Convolutional Neural Networks (CNNs) are artificial learning systems typically based on two operations: convolution, which implements feature extraction through filtering, and pooling, which implements dimensionality reduction. The impact of pooling in the classification performance of the CNNs has been highlighted in several previous works, and a variety of alternative pooling operators have been proposed. However, only a few of them tackle with the uncertainty that is naturally propagated from the input layer to the feature maps of the hidden layers through convolutions. In this paper we pre…
University of New South Wales（新南威尔士大学）；Max Planck Institute for Mathematics in the Sciences（马克斯·普朗克数学科学研究所）；Zhejiang Normal University（浙江师范大学）；Princeton University（普林斯顿大学）；University of California, Los Angeles（加州大学洛杉矶分校）；City University of Hong Kong（香港城市大学）
Deep Graph Neural Networks (GNNs) are useful models for graph classification and graph-based regression tasks. In these tasks, graph pooling is a critical ingredient by which GNNs adapt to input graphs of varying size and structure. We propose a new graph pooling operation based on compressive Haar transforms -- HaarPooling. HaarPooling implements a cascade of pooling operations; it is computed by following a sequence of clusterings of the input graph. A HaarPooling layer transforms a given input graph to an output graph with a smaller node number and the same feature dimension; the compressiv…
Shahrood University of Technology（沙赫鲁德工业大学）；Faculty of Electrical & Robotics Engineering（电气与机器人工程学院）
Nowadays, Deep Neural Networks are among the main tools used in various sciences. Convolutional Neural Network is a special type of DNN consisting of several convolution layers, each followed by an activation function and a pooling layer. The pooling layer is an important layer that executes the down-sampling on the feature maps coming from the previous layer and produces new feature maps with a condensed resolution. This layer drastically reduces the spatial dimension of input. It serves two main purposes. The first is to reduce the number of parameters or weights, thus lessening the computat…
ENADPool: The Edge-Node Attention-based Differentiable Pooling for Graph Neural Networks
School of Artificial Intelligence, Beijing Normal University（北京师范大学人工智能学院）；School of Information, Central University of Finance and Economics（中央财经大学信息学院）；Key Laboratory of Intelligent Education Technology and Application of Zhejiang Province, Zhejiang Normal University（浙江师范大学浙江省智能教育技术与应用重点实验室）；Department of Computer Science, University of York（约克大学计算机科学系）
Duke University（杜克大学）；NEC Laboratories America（美国电气通信技术公司（NEC）实验室）；Sun Yat-sen University（中山大学）；Microsoft Research（微软研究院）
Many deep learning architectures have been proposed to model the compositionality in text sequences, requiring a substantial number of parameters and expensive computations. However, there has not been a rigorous evaluation regarding the added value of sophisticated compositional functions. In this paper, we conduct a point-by-point comparative study between Simple Word-Embedding-based Models (SWEMs), consisting of parameter-free pooling operations, relative to word-embedding-based RNN/CNN models. Surprisingly, SWEMs exhibit comparable or even superior performance in the majority of cases cons…
Despite its high search efficiency, differential architecture search (DARTS) often selects network architectures with dominated skip connections which lead to performance degradation. However, theoretical understandings on this issue remain absent, hindering the development of more advanced methods in a principled way. In this work, we solve this problem by theoretically analyzing the effects of various types of operations, e.g. convolution, skip connection and zero operation, to the network optimization. We prove that the architectures with more skip connections can converge faster than the o…
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…
Temple University（天普大学）；University of Texas at Dallas（德克萨斯大学达拉斯分校）
Graph neural networks (GNNs) have demonstrated a significant success in various graph learning tasks, from graph classification to anomaly detection. There recently has emerged a number of approaches adopting a graph pooling operation within GNNs, with a goal to preserve graph attributive and structural features during the graph representation learning. However, most existing graph pooling operations suffer from the limitations of relying on node-wise neighbor weighting and embedding, which leads to insufficient encoding of rich topological structures and node attributes exhibited by real-worl…
Deep learning methods for graphs have seen rapid progress in recent years with much focus awarded to generalising Convolutional Neural Networks (CNN) to graph data. CNNs are typically realised by alternating convolutional and pooling layers where the pooling layers subsample the grid and exchange spatial or temporal resolution for increased feature dimensionality. Whereas the generalised convolution operator for graphs has been studied extensively and proven useful, hierarchical coarsening of graphs is still challenging since nodes in graphs have no spatial locality and no natural order. This …
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…
Iowa State University（爱荷华州立大学）；Texas A&M University（得克萨斯农工大学）
Pooling operations have shown to be effective on computer vision and natural language processing tasks. One challenge of performing pooling operations on graph data is the lack of locality that is not well-defined on graphs. Previous studies used global ranking methods to sample some of the important nodes, but most of them are not able to incorporate graph topology. In this work, we propose the topology-aware pooling (TAP) layer that explicitly considers graph topology. Our TAP layer is a two-stage voting process that selects more important nodes in a graph. It first performs local voting to …
University of Saskatchewan（萨斯喀彻温大学）；University of Tulane（杜兰大学）
Breast cancer is one of the most common causes of cancer-related death in women worldwide. Early and accurate diagnosis of breast cancer may significantly increase the survival rate of patients. In this study, we aim to develop a fully automatic, deep learning-based, method using descriptor features extracted by Deep Convolutional Neural Network (DCNN) models and pooling operation for the classification of hematoxylin and eosin stain (H&E) histological breast cancer images provided as a part of the International Conference on Image Analysis and Recognition (ICIAR) 2018 Grand Challenge on BreAs…
Beyond Convolution: Advancing Hypergraph Neural Networks with Hypergraph U-Nets
Institute for Financial Services Analytics, University of Delaware（特拉华大学金融服务分析研究所）；Department of Applied Economics and Statistics, University of Delaware（特拉华大学应用经济学与统计学系）；Department of Electrical and Computer Engineering, University of Kentucky（肯塔基大学电气与计算机工程系）；Department of Electrical and Computer Engineering, University of Delaware（特拉华大学电气与计算机工程系）
Geometry-Aware Edge Pooling for Graph Neural Networks
Helmholtz Munich（海德堡-穆恩大学）；Technical University of Munich（慕尼黑技术大学）；Université de Montréal（蒙特利尔大学）；Mila - Quebec AI Institute（魁北克人工智能研究所）；Université de Fribourg（弗里堡大学）
Korea Advanced Institute of Science and Technology（韩国科学技术院）；Mokwon University（牧园大学）
Convolutional neural networks (CNNs) with convolutional and pooling operations along the frequency axis have been proposed to attain invariance to frequency shifts of features. However, this is inappropriate with regard to the fact that acoustic features vary in frequency. In this paper, we contend that convolution along the time axis is more effective. We also propose the addition of an intermap pooling (IMP) layer to deep CNNs. In this layer, filters in each group extract common but spectrally variant features, then the layer pools the feature maps of each group. As a result, the proposed IM…
College of Science and Engineering, University of Minnesota（明尼苏达大学科学与工程学院）
We propose a new network architecture for standard spin-Hall magnetic tunnel junction-based spintronic neurons that allows them to compute multiple critical convolutional neural network functionalities simultaneously and in parallel, saving space and time. An approximation to the Rectified Linear Unit transfer function and the local pooling function are computed simultaneously with the convolution operation itself. A proof-of-concept simulation is performed on the MNIST dataset, achieving up to 98% accuracy at a cost of less than 1 nJ for all convolution, activation and pooling operations comb…
Similarity measures based purely on word embeddings are comfortably competing with much more sophisticated deep learning and expert-engineered systems on unsupervised semantic textual similarity (STS) tasks. In contrast to commonly used geometric approaches, we treat a single word embedding as e.g. 300 observations from a scalar random variable. Using this paradigm, we first illustrate that similarities derived from elementary pooling operations and classic correlation coefficients yield excellent results on standard STS benchmarks, outperforming many recently proposed methods while being much…
