We study the expressive power of deep polynomial neural networks through the geometry of their neurovariety. We introduce the notion of the activation degree threshold of a network architecture to express when the dimension of the neurovariety achieves its theoretical maximum. We prove the existence of the activation degree threshold for all polynomial neural networks without width-one bottlenecks and demonstrate a universal upper bound that is quadratic in the width of largest size. In doing so, we prove the high activation degree conjecture of Kileel, Trager, and Bruna. Certain structured architectures have exceptional activation degree thresholds, making them especially expressive in the sense of their neurovariety dimension. In this direction, we prove that polynomial neural networks with equi-width architectures are maximally expressive by showing their activation degree threshold is one.
Rectified Linear Units (ReLUs) have been shown to ameliorate the vanishing gradient problem, allow for efficient backpropagation, and empirically promote sparsity in the learned parameters. They have led to state-of-the-art results in a variety of applications. However, unlike threshold and sigmoid networks, ReLU networks are less explored from the perspective of their expressiveness. This paper studies the expressiveness of ReLU networks. We characterize the decision boundary of two-layer ReLU networks by constructing functionally equivalent threshold networks. We show that while the decision…
Model compression has become necessary when applying neural networks (NN) into many real application tasks that can accept slightly-reduced model accuracy with strict tolerance to model complexity. Recently, Knowledge Distillation, which distills the knowledge from well-trained and highly complex teacher model into a compact student model, has been widely used for model compression. However, under the strict requirement on the resource cost, it is quite challenging to achieve comparable performance with the teacher model, essentially due to the drastically-reduced expressiveness ability of the…
University of Toronto（多伦多大学）；Vector Institute for Artificial Intelligence（向量人工智能研究所）
Deep neural networks (DNNs) have emerged as a popular mathematical tool for function approximation due to their capability of modelling highly nonlinear functions. Their applications range from image classification and natural language processing to learning-based control. Despite their empirical successes, there is still a lack of theoretical understanding of the representative power of such deep architectures. In this work, we provide a theoretical analysis of the expressiveness of fully-connected, feedforward DNNs with 1-Lipschitz activation functions. In particular, we characterize the exp…
Overwhelming theoretical and empirical evidence shows that mildly overparametrized neural networks -- those with more connections than the size of the training data -- are often able to memorize the training data with $100\%$ accuracy. This was rigorously proved for networks with sigmoid activation functions and, very recently, for ReLU activations. Addressing a 1988 open question of Baum, we prove that this phenomenon holds for general multilayered perceptrons, i.e. neural networks with threshold activation functions, or with any mix of threshold and ReLU activations. Our construction is prob…
Threshold activation functions are highly preferable in neural networks due to their efficiency in hardware implementations. Moreover, their mode of operation is more interpretable and resembles that of biological neurons. However, traditional gradient based algorithms such as Gradient Descent cannot be used to train the parameters of neural networks with threshold activations since the activation function has zero gradient except at a single non-differentiable point. To this end, we study weight decay regularized training problems of deep neural networks with threshold activations. We first s…
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, …
University of Texas at Austin（得克萨斯大学奥斯汀分校）；Google（谷歌）
Giving provable guarantees for learning neural networks is a core challenge of machine learning theory. Most prior work gives parameter recovery guarantees for one hidden layer networks, however, the networks used in practice have multiple non-linear layers. In this work, we show how we can strengthen such results to deeper networks -- we address the problem of uncovering the lowest layer in a deep neural network under the assumption that the lowest layer uses a high threshold before applying the activation, the upper network can be modeled as a well-behaved polynomial and the input distributi…
University of Antwerp（安特卫普大学）；Hasselt University（哈塞尔特大学）
We investigate the power of message-passing neural networks (MPNNs) in their capacity to transform the numerical features stored in the nodes of their input graphs. Our focus is on global expressive power, uniformly over all input graphs, or over graphs of bounded degree with features from a bounded domain. Accordingly, we introduce the notion of a global feature map transformer (GFMT). As a yardstick for expressiveness, we use a basic language for GFMTs, which we call MPLang. Every MPNN can be expressed in MPLang, and our results clarify to which extent the converse inclusion holds. We consid…
Efficient Neural Networks with Discrete Cosine Transform Activations
Centre Tecnològic de Telecomunicacions de Catalunya, Spain（西班牙加泰罗尼亚电信技术中心）；Dept. of Signal Theory and Communications, Universitat Politècnica de Catalunya, Spain（西班牙加泰罗尼亚理工大学信号理论与通信系）；ICREA Acadèmia, Spain（西班牙ICREA学术机构）
Princeton University（普林斯顿大学）；New York University（纽约大学）
We study deep neural networks with polynomial activations, particularly their expressive power. For a fixed architecture and activation degree, a polynomial neural network defines an algebraic map from weights to polynomials. The image of this map is the functional space associated to the network, and it is an irreducible algebraic variety upon taking closure. This paper proposes the dimension of this variety as a precise measure of the expressive power of polynomial neural networks. We obtain several theoretical results regarding this dimension as a function of architecture, including an exac…
A Logical View of GNN-Style Computation and the Role of Activation Functions
Institute for Mathematical and Computational Engineering（数学与计算工程研究所）；Pontifical Catholic University of Chile（天主教智利大学）；IMFD；CENIA；Department of Computer Science, University of Antwerp（安特卫普大学计算机科学系）；Institute for Logic and Computation, TU Wien（逻辑与计算研究所，维也纳技术大学）；Data Science Institute, Universiteit Hasselt（数据科学研究所，哈塞尔特大学）
University of Siena（锡耶纳大学）；University of Florence（佛罗伦萨大学）
The effectiveness of deep neural architectures has been widely supported in terms of both experimental and foundational principles. There is also clear evidence that the activation function (e.g. the rectifier and the LSTM units) plays a crucial role in the complexity of learning. Based on this remark, this paper discusses an optimal selection of the neuron non-linearity in a functional framework that is inspired from classic regularization arguments. It is shown that the best activation function is represented by a kernel expansion in the training set, that can be effectively approximated ove…
Complex-Weighted Convolutional Networks: Provable Expressiveness via Complex Diffusion
IST Austria（IST奥地利研究院）；Center for Systems Biology Dresden（德累斯顿系统生物学中心）；Mathematical bioPhysics Group, Max Planck Institute for Multidisciplinary Sciences（多学科科学研究所数学生物物理组）；Max Planck Institute of Molecular Cell Biology and Genetics（分子细胞生物学和遗传学马克斯·普朗克研究所）；Max Planck Institute for the Physics of Complex Systems（复杂系统物理马克斯·普朗克研究所）；Dresden University of Technology（德累斯顿技术大学）
Pohang University of Science and Technology (POSTECH)（浦项科技大学）
Binarization of neural network models is considered as one of the promising methods to deploy deep neural network models on resource-constrained environments such as mobile devices. However, Binary Neural Networks (BNNs) tend to suffer from severe accuracy degradation compared to the full-precision counterpart model. Several techniques were proposed to improve the accuracy of BNNs. One of the approaches is to balance the distribution of binary activations so that the amount of information in the binary activations becomes maximum. Based on extensive analysis, in stark contrast to previous work…
University of Oulu（奥卢大学）；National University of Defense Technology（国防科技大学）
Binary neural networks (BNNs) constrain weights and activations to +1 or -1 with limited storage and computational cost, which is hardware-friendly for portable devices. Recently, BNNs have achieved remarkable progress and been adopted into various fields. However, the performance of BNNs is sensitive to activation distribution. The existing BNNs utilized the Sign function with predefined or learned static thresholds to binarize activations. This process limits representation capacity of BNNs since different samples may adapt to unequal thresholds. To address this problem, we propose a dynamic…
Bio-inspired spiking neural networks (SNNs), operating with asynchronous binary signals (or spikes) distributed over time, can potentially lead to greater computational efficiency on event-driven hardware. The state-of-the-art SNNs suffer from high inference latency, resulting from inefficient input encoding, and sub-optimal settings of the neuron parameters (firing threshold, and membrane leak). We propose DIET-SNN, a low-latency deep spiking network that is trained with gradient descent to optimize the membrane leak and the firing threshold along with other network parameters (weights). The …
Department of Electrical Engineering, Tsinghua University（清华大学电子工程系）
The developments of deep neural networks (DNN) in recent years have ushered a brand new era of artificial intelligence. DNNs are proved to be excellent in solving very complex problems, e.g., visual recognition and text understanding, to the extent of competing with or even surpassing people. Despite inspiring and encouraging success of DNNs, thorough theoretical analyses still lack to unravel the mystery of their magics. The design of DNN structure is dominated by empirical results in terms of network depth, number of neurons and activations. A few of remarkable works published recently in an…
fortiss - Research Institute of the Free State of Bavaria（fortiss——巴伐利亚自由州研究院）；DENSO CORPORATION（电装公司）
For using neural networks in safety critical domains, it is important to know if a decision made by a neural network is supported by prior similarities in training. We propose runtime neuron activation pattern monitoring - after the standard training process, one creates a monitor by feeding the training data to the network again in order to store the neuron activation patterns in abstract form. In operation, a classification decision over an input is further supplemented by examining if a pattern similar (measured by Hamming distance) to the generated pattern is contained in the monitor. If t…
Bucknell University（巴克内尔大学）；The University of Utah（犹他大学）
We can compare the expressiveness of neural networks that use rectified linear units (ReLUs) by the number of linear regions, which reflect the number of pieces of the piecewise linear functions modeled by such networks. However, enumerating these regions is prohibitive and the known analytical bounds are identical for networks with same dimensions. In this work, we approximate the number of linear regions through empirical bounds based on features of the trained network and probabilistic inference. Our first contribution is a method to sample the activation patterns defined by ReLUs using uni…
Moscow Institute of Physics and Technology（莫斯科物理技术学院）
We prove that if an activation function satisfies some mild conditions and number of neurons in a two-layered fully connected neural network with this activation function is beyond a certain threshold, then gradient descent on quadratic loss function finds the optimal weights of input layer for global minima in linear time. This threshold value is an improvement over previously obtained values. We hypothesise that this bound cannot be improved by the method we are using in this work.
We address the problem of reconstructing sparse signals from noisy and compressive measurements using a feed-forward deep neural network (DNN) with an architecture motivated by the iterative shrinkage-thresholding algorithm (ISTA). We maintain the weights and biases of the network links as prescribed by ISTA and model the nonlinear activation function using a linear expansion of thresholds (LET), which has been very successful in image denoising and deconvolution. The optimal set of coefficients of the parametrized activation is learned over a training dataset containing measurement-sparse sig…
TU Delft（代尔夫特理工大学）；Shell Global Solutions International B.V.（壳牌全球解决方案国际有限公司）
Binary Neural Networks (BNNs) are receiving an upsurge of attention for bringing power-hungry deep learning towards edge devices. The traditional wisdom in this space is to employ sign() for binarizing featuremaps. We argue and illustrate that sign() is a uniqueness bottleneck, limiting information propagation throughout the network. To alleviate this, we propose to dispense sign(), replacing it with a learnable activation binarizer (LAB), allowing the network to learn a fine-grained binarization kernel per layer - as opposed to global thresholding. LAB is a novel universal module that can sea…
