We consider the problem of training a deep orthogonal linear network, which consists of a product of orthogonal matrices, with no non-linearity in-between. We show that training the weights with Riemannian gradient descent is equivalent to training the whole factorization by gradient descent. This means that there is no effect of overparametrization and implicit bias at all in this setting: training such a deep, overparametrized, network is perfectly equivalent to training a one-layer shallow network.
Center for Data Science, New York University（纽约大学数据科学中心）；Inria - Département d’Informatique de l’École Normale Supérieure（法国国家信息与自动化研究所 - 巴黎高等师范学校计算机科学系）；PSL Research University（巴黎文理研究大学）
Deep networks are often considered to be more expressive than shallow ones in terms of approximation. Indeed, certain functions can be approximated by deep networks provably more efficiently than by shallow ones, however, no tractable algorithms are known for learning such deep models. Separately, a recent line of work has shown that deep networks trained with gradient descent may behave like (tractable) kernel methods in a certain over-parameterized regime, where the kernel is determined by the architecture and initialization, and this paper focuses on approximation for such kernels. We show …
School of Computer Science, The Hebrew University（计算机科学学院，希伯来大学）
Understanding the power of depth in feed-forward neural networks is an ongoing challenge in the field of deep learning theory. While current works account for the importance of depth for the expressive power of neural-networks, it remains an open question whether these benefits are exploited during a gradient-based optimization process. In this work we explore the relation between expressivity properties of deep networks and the ability to train them efficiently using gradient-based algorithms. We give a depth separation argument for distributions with fractal structure, showing that they can …
University of Strathclyde（斯特拉斯克莱德大学）；University of Abertay Dundee（阿伯泰邓迪大学）
Intrusion detection has attracted a considerable interest from researchers and industries. The community, after many years of research, still faces the problem of building reliable and efficient IDS that are capable of handling large quantities of data, with changing patterns in real time situations. The work presented in this manuscript classifies intrusion detection systems (IDS). Moreover, a taxonomy and survey of shallow and deep networks intrusion detection systems is presented based on previous and current works. This taxonomy and survey reviews machine learning techniques and their perf…
University of Maryland, College Park（马里兰大学帕克分校）；NVIDIA Corporation（英伟达公司）
Enforcing orthogonality in neural networks is an antidote for gradient vanishing/exploding problems, sensitivity by adversarial perturbation, and bounding generalization errors. However, many previous approaches are heuristic, and the orthogonality of convolutional layers is not systematically studied: some of these designs are not exactly orthogonal, while others only consider standard convolutional layers and propose specific classes of their realizations. To address this problem, we propose a theoretical framework for orthogonal convolutional layers, which establishes the equivalence betwee…
The University of Western Australia（西澳大利亚大学）；The University of New South Wales（新南威尔士大学）
In this paper, we introduce transformations of deep rectifier networks, enabling the conversion of deep rectifier networks into shallow rectifier networks. We subsequently prove that any rectifier net of any depth can be represented by a maximum of a number of functions that can be realized by a shallow network with a single hidden layer. The transformations of both deep rectifier nets and deep residual nets are conducted to demonstrate the advantages of the residual nets over the conventional neural nets and the advantages of the deep neural nets over the shallow neural nets. In summary, for …
University of California, Los Angeles（加州大学洛杉矶分校）；DeepMind（深度思考（DeepMind））；Adobe Research（Adobe研究院）
We study a general class of contextual bandits, where each context-action pair is associated with a raw feature vector, but the reward generating function is unknown. We propose a novel learning algorithm that transforms the raw feature vector using the last hidden layer of a deep ReLU neural network (deep representation learning), and uses an upper confidence bound (UCB) approach to explore in the last linear layer (shallow exploration). We prove that under standard assumptions, our proposed algorithm achieves $\tilde{O}(\sqrt{T})$ finite-time regret, where $T$ is the learning time horizon. C…
RJ Research Consulting（RJ研究咨询公司）；Rutgers University（罗格斯大学）
This paper reports the performances of shallow word-level convolutional neural networks (CNN), our earlier work (2015), on the eight datasets with relatively large training data that were used for testing the very deep character-level CNN in Conneau et al. (2016). Our findings are as follows. The shallow word-level CNNs achieve better error rates than the error rates reported in Conneau et al., though the results should be interpreted with some consideration due to the unique pre-processing of Conneau et al. The shallow word-level CNN uses more parameters and therefore requires more storage th…
We prove Carl's type inequalities for the error of approximation of compact sets K by deep and shallow neural networks. This in turn gives lower bounds on how well we can approximate the functions in K when requiring the approximants to come from outputs of such networks. Our results are obtained as a byproduct of the study of the recently introduced Lipschitz widths.
MINES ParisTech, PSL Research University（巴黎高等矿业学院，PSL研究大学）；Institut Curie, PSL Research University（居里研究所，PSL研究大学）；INSERM（法国国家健康与医学研究院）；Google Brain（谷歌大脑）
Dropout is a regularisation technique in neural network training where unit activations are randomly set to zero with a given probability \emph{independently}. In this work, we propose a generalisation of dropout and other multiplicative noise injection schemes for shallow and deep neural networks, where the random noise applied to different units is not independent but follows a joint distribution that is either fixed or estimated during training. We provide theoretical insights on why such adaptive structured noise injection (ASNI) may be relevant, and empirically confirm that it helps boost…
Department of Statistical Sciences, University of Bologna, Italy（博洛尼亚大学统计科学系）
This paper focuses on a comparative evaluation of the most common and modern methods for text classification, including the recent deep learning strategies and ensemble methods. The study is motivated by a challenging real data problem, characterized by high-dimensional and extremely sparse data, deriving from incoming calls to the customer care of an Italian phone company. We will show that deep learning outperforms many classical (shallow) strategies but the combination of shallow and deep learning methods in a unique ensemble classifier may improve the robustness and the accuracy of "single…
This paper considers the power of deep neural networks (deep nets for short) in realizing data features. Based on refined covering number estimates, we find that, to realize some complex data features, deep nets can improve the performances of shallow neural networks (shallow nets for short) without requiring additional capacity costs. This verifies the advantage of deep nets in realizing complex features. On the other hand, to realize some simple data feature like the smoothness, we prove that, up to a logarithmic factor, the approximation rate of deep nets is asymptotically identical to that…
We show that the output of a (residual) convolutional neural network (CNN) with an appropriate prior over the weights and biases is a Gaussian process (GP) in the limit of infinitely many convolutional filters, extending similar results for dense networks. For a CNN, the equivalent kernel can be computed exactly and, unlike "deep kernels", has very few parameters: only the hyperparameters of the original CNN. Further, we show that this kernel has two properties that allow it to be computed efficiently; the cost of evaluating the kernel for a pair of images is similar to a single forward pass t…
The Chinese University of Hong Kong（香港中文大学）；Dalian University of Technology（大连理工大学）
VLSI mask optimization is one of the most critical stages in manufacturability aware design, which is costly due to the complicated mask optimization and lithography simulation. Recent researches have shown prominent advantages of machine learning techniques dealing with complicated and big data problems, which bring potential of dedicated machine learning solution for DFM problems and facilitate the VLSI design cycle. In this paper, we focus on a heterogeneous OPC framework that assists mask layout optimization. Preliminary results show the efficiency and effectiveness of proposed frameworks …
The University of Iowa（爱荷华大学）；University of Central Florida（中佛罗里达大学）
Dropout has been witnessed with great success in training deep neural networks by independently zeroing out the outputs of neurons at random. It has also received a surge of interest for shallow learning, e.g., logistic regression. However, the independent sampling for dropout could be suboptimal for the sake of convergence. In this paper, we propose to use multinomial sampling for dropout, i.e., sampling features or neurons according to a multinomial distribution with different probabilities for different features/neurons. To exhibit the optimal dropout probabilities, we analyze the shallow l…
Florida State University（佛罗里达州立大学）；University of Chinese Academy of Sciences（中国科学院大学）
In this paper, we aim at introducing a new machine learning model, namely reconciled polynomial machine, which can provide a unified representation of existing shallow and deep machine learning models. Reconciled polynomial machine predicts the output by computing the inner product of the feature kernel function and variable reconciling function. Analysis of several concrete models, including Linear Models, FM, MVM, Perceptron, MLP and Deep Neural Networks, will be provided in this paper, which can all be reduced to the reconciled polynomial machine representations. Detailed analysis of the le…
Deakin University（迪肯大学）；Applied Artificial Intelligence Institute (A2I2)（应用人工智能研究所（A2I2））
In this paper we explore a connection between deep networks and learning in reproducing kernel Krein space. Our approach is based on the concept of push-forward - that is, taking a fixed non-linear transform on a linear projection and converting it to a linear projection on the output of a fixed non-linear transform, pushing the weights forward through the non-linearity. Applying this repeatedly from the input to the output of a deep network, the weights can be progressively "pushed" to the output layer, resulting in a flat network that has the form of a fixed non-linear map (whose form is det…
Ecole Polytechnique Fédérale de Lausanne（洛桑联邦理工学院）；University of Lausanne（洛桑大学）；Siemens Healthcare AG（西门子医疗股份公司）；Lausanne University Hospital（洛桑大学医院）；University Hospital Basel（巴塞尔大学医院）；University of Basel（巴塞尔大学）
In this work, we present a comparison of a shallow and a deep learning architecture for the automated segmentation of white matter lesions in MR images of multiple sclerosis patients. In particular, we train and test both methods on early stage disease patients, to verify their performance in challenging conditions, more similar to a clinical setting than what is typically provided in multiple sclerosis segmentation challenges. Furthermore, we evaluate a prototype naive combination of the two methods, which refines the final segmentation. All methods were trained on 32 patients, and the evalua…
University of St.Gallen（圣加仑大学）；University Passau（帕绍大学）
There is some theoretical evidence that deep neural networks with multiple hidden layers have a potential for more efficient representation of multidimensional mappings than shallow networks with a single hidden layer. The question is whether it is possible to exploit this theoretical advantage for finding such representations with help of numerical training methods. Tests using prototypical problems with a known mean square minimum did not confirm this hypothesis. Minima found with the help of deep networks have always been worse than those found using shallow networks. This does not directly…
Institute of Mathematical Sciences, Claremont Graduate University（克莱蒙特研究生大学数学科学研究所）；Center for Brains, Minds, and Machines, McGovern Institute for Brain Research, Massachusetts Institute of Technology（麻省理工学院麦戈文脑科学研究所脑、心智与机器中心）
We show that deep networks are better than shallow networks at approximating functions that can be expressed as a composition of functions described by a directed acyclic graph, because the deep networks can be designed to have the same compositional structure, while a shallow network cannot exploit this knowledge. Thus, the blessing of compositionality mitigates the curse of dimensionality. On the other hand, a theorem called good propagation of errors allows to `lift' theorems about shallow networks to those about deep networks with an appropriate choice of norms, smoothness, etc. We illustr…
Institute of Mathematical Sciences, Claremont Graduate University（克莱蒙特研究生大学数学科学研究所）
This paper proves an abstract theorem addressing in a unified manner two important problems in function approximation: avoiding curse of dimensionality and estimating the degree of approximation for out-of-sample extension in manifold learning. We consider an abstract (shallow) network that includes, for example, neural networks, radial basis function networks, and kernels on data defined manifolds used for function approximation in various settings. A deep network is obtained by a composition of the shallow networks according to a directed acyclic graph, representing the architecture of the d…
Center for Computational Engineering and Networking (CEN)（计算工程与网络中心（CEN））；Amrita School of Engineering（阿姆里塔工程学院）；Amrita Vishwa Vidyapeetham（阿姆里塔维什瓦维迪亚佩瑟姆大学）
The knowledge regarding the function of proteins is necessary as it gives a clear picture of biological processes. Nevertheless, there are many protein sequences found and added to the databases but lacks functional annotation. The laboratory experiments take a considerable amount of time for annotation of the sequences. This arises the need to use computational techniques to classify proteins based on their functions. In our work, we have collected the data from Swiss-Prot containing 40433 proteins which is grouped into 30 families. We pass it to recurrent neural network(RNN), long short term…
Stanford University（斯坦福大学）；University of Wisconsin-Madison（威斯康星大学麦迪逊分校）
We present Caffe con Troll (CcT), a fully compatible end-to-end version of the popular framework Caffe with rebuilt internals. We built CcT to examine the performance characteristics of training and deploying general-purpose convolutional neural networks across different hardware architectures. We find that, by employing standard batching optimizations for CPU training, we achieve a 4.5x throughput improvement over Caffe on popular networks like CaffeNet. Moreover, with these improvements, the end-to-end training time for CNNs is directly proportional to the FLOPS delivered by the CPU, which e…
Perimeter Institute for Theoretical Physics（理论物理圆周研究所）；University of Waterloo（滑铁卢大学）
It is well established that neural networks with deep architectures perform better than shallow networks for many tasks in machine learning. In statistical physics, while there has been recent interest in representing physical data with generative modelling, the focus has been on shallow neural networks. A natural question to ask is whether deep neural networks hold any advantage over shallow networks in representing such data. We investigate this question by using unsupervised, generative graphical models to learn the probability distribution of a two-dimensional Ising system. Deep Boltzmann …
NEC Labs America（NEC美国实验室）；National Research Council Canada（加拿大国家研究委员会）
Explicit high-order feature interactions efficiently capture essential structural knowledge about the data of interest and have been used for constructing generative models. We present a supervised discriminative High-Order Parametric Embedding (HOPE) approach to data visualization and compression. Compared to deep embedding models with complicated deep architectures, HOPE generates more effective high-order feature mapping through an embarrassingly simple shallow model. Furthermore, two approaches to generating a small number of exemplars conveying high-order interactions to represent large-s…
Singapore University of Technology and Design (SUTD)（新加坡科技设计大学）；Thales Solutions Asia Pte Ltd（泰雷兹亚洲解决方案私人有限公司）
Shallow water environments create a challenging channel for communications. In this paper, we focus on the challenges posed by the frequency-selective signal distortion called the Doppler effect. We explore the design and performance of machine learning (ML) based demodulation methods --- (1) Deep Belief Network-feed forward Neural Network (DBN-NN) and (2) Deep Belief Network-Convolutional Neural Network (DBN-CNN) in the physical layer of Shallow Water Acoustic Communication (SWAC). The proposed method comprises of a ML based feature extraction method and classification technique. First, the f…
We characterize a prevalent weakness of deep neural networks (DNNs)---overthinking---which occurs when a DNN can reach correct predictions before its final layer. Overthinking is computationally wasteful, and it can also be destructive when, by the final layer, a correct prediction changes into a misclassification. Understanding overthinking requires studying how each prediction evolves during a DNN's forward pass, which conventionally is opaque. For prediction transparency, we propose the Shallow-Deep Network (SDN), a generic modification to off-the-shelf DNNs that introduces internal classif…
This paper explores a new framework for lossy image encryption and decryption using a simple shallow encoder neural network E for encryption, and a complex deep decoder neural network D for decryption. E is kept simple so that encoding can be done on low power and portable devices and can in principle be any nonlinear function which outputs an encoded vector. D is trained to decode the encodings using the dataset of image - encoded vector pairs obtained from E and happens independently of E. As the encodings come from E which while being a simple neural network, still has thousands of random p…
Training deep neural networks with the error backpropagation algorithm is considered implausible from a biological perspective. Numerous recent publications suggest elaborate models for biologically plausible variants of deep learning, typically defining success as reaching around 98% test accuracy on the MNIST data set. Here, we investigate how far we can go on digit (MNIST) and object (CIFAR10) classification with biologically plausible, local learning rules in a network with one hidden layer and a single readout layer. The hidden layer weights are either fixed (random or random Gabor filter…
We bound the excess risk of interpolating deep linear networks trained using gradient flow. In a setting previously used to establish risk bounds for the minimum $\ell_2$-norm interpolant, we show that randomly initialized deep linear networks can closely approximate or even match known bounds for the minimum $\ell_2$-norm interpolant. Our analysis also reveals that interpolating deep linear models have exactly the same conditional variance as the minimum $\ell_2$-norm solution. Since the noise affects the excess risk only through the conditional variance, this implies that depth does not impr…
Technische Universität Berlin（柏林工业大学）；Fraunhofer Heinrich Hertz Institute（弗劳恩霍夫海因里希·赫兹研究所）；Technische Universität Kaiserslautern（凯泽斯劳滕工业大学）；Oregon State University（俄勒冈州立大学）；Google Research（谷歌研究院）；Korea University（高丽大学）；Max Planck Institute for Informatics（马克斯·普朗克信息学研究所）
Deep learning approaches to anomaly detection have recently improved the state of the art in detection performance on complex datasets such as large collections of images or text. These results have sparked a renewed interest in the anomaly detection problem and led to the introduction of a great variety of new methods. With the emergence of numerous such methods, including approaches based on generative models, one-class classification, and reconstruction, there is a growing need to bring methods of this field into a systematic and unified perspective. In this review we aim to identify the co…
How deep is your network? Deep vs. shallow learning of transfer operators
Maxwell Institute for Mathematical Sciences, University of Edinburgh and Heriot–Watt University（数学科学研究所、爱丁堡大学和赫里奥特-沃斯大学）；School of Mathematics, University of Edinburgh（数学学院、爱丁堡大学）；School of Mathematical and Computer Sciences, Heriot–Watt University（数学与计算机科学学院、赫里奥特-沃斯大学）
The University of Hong Kong（香港大学）；The University of Texas at Austin（得克萨斯大学奥斯汀分校）
Modern machine learning algorithms have been adopted in a range of signal-processing applications spanning computer vision, natural language processing, and artificial intelligence. Many relevant problems involve subspace-structured features, orthogonality constrained or low-rank constrained objective functions, or subspace distances. These mathematical characteristics are expressed naturally using the Grassmann manifold. Unfortunately, this fact is not yet explored in many traditional learning algorithms. In the last few years, there have been growing interests in studying Grassmann manifold …
In the past, Acoustic Scene Classification systems have been based on hand crafting audio features that are input to a classifier. Nowadays, the common trend is to adopt data driven techniques, e.g., deep learning, where audio representations are learned from data. In this paper, we propose a system that consists of a simple fusion of two methods of the aforementioned types: a deep learning approach where log-scaled mel-spectrograms are input to a convolutional neural network, and a feature engineering approach, where a collection of hand-crafted features is input to a gradient boosting machin…
