Low-Rank Learning by Design: the Role of Network Architecture and Activation Linearity in Gradient Rank Collapse
Translational Research Center for Neuroimaging and Data Science (TReNDs)（神经成像与数据科学转化研究中心（TReNDs））；Georgia State University（佐治亚州立大学）；Georgia Institute of Technology（佐治亚理工学院）；Emory University（埃默里大学）；Maynooth University（梅努斯大学）
Bradley T. Baker, Barak A. Pearlmutter, Robyn Miller, Vince D. Calhoun, Sergey M. Plis
Our understanding of learning dynamics of deep neural networks (DNNs) remains incomplete. Recent research has begun to uncover the mathematical principles underlying these networks, including the phenomenon of "Neural Collapse", where linear classifiers within DNNs converge to specific geometrical structures during late-stage training. However, the role of geometric constraints in learning extends beyond this terminal phase. For instance, gradients in fully-connected layers naturally develop a low-rank structure due to the accumulation of rank-one outer products over a training batch. Despite the attention given to methods that exploit this structure for memory saving or regularization, the emergence of low-rank learning as an inherent aspect of certain DNN architectures has been under-explored. In this paper, we conduct a comprehensive study of gradient rank in DNNs, examining how architectural choices and structure of the data effect gradient rank bounds. Our theoretical analysis provides these bounds for training fully-connected, recurrent, and convolutional neural networks. We also demonstrate, both theoretically and empirically, how design choices like activation function linearity, bottleneck layer introduction, convolutional stride, and sequence truncation influence these bounds. Our findings not only contribute to the understanding of learning dynamics in DNNs, but also provide practical guidance for deep learning engineers to make informed design decisions.
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…
Certified defenses against small-norm adversarial examples have received growing attention in recent years; though certified accuracies of state-of-the-art methods remain far below their non-robust counterparts, despite the fact that benchmark datasets have been shown to be well-separated at far larger radii than the literature generally attempts to certify. In this work, we offer insights that identify potential factors in this performance gap. Specifically, our analysis reveals that piecewise linearity imposes fundamental limitations on the tightness of leading certification techniques. Thes…
University of British Columbia（不列颠哥伦比亚大学）；Microsoft Research（微软研究院）
Echo State Networks (ESNs) are a special type of the temporally deep network model, the Recurrent Neural Network (RNN), where the recurrent matrix is carefully designed and both the recurrent and input matrices are fixed. An ESN uses the linearity of the activation function of the output units to simplify the learning of the output matrix. In this paper, we devise a special technique that take advantage of this linearity in the output units of an ESN, to learn the input and recurrent matrices. This has not been done in earlier ESNs due to their well known difficulty in learning those matrices.…
The reason behind CNNs capability to learn high-dimensional complex features from the images is the non-linearity introduced by the activation function. Several advanced activation functions have been discovered to improve the training process of neural networks, as choosing an activation function is a crucial step in the modeling. Recent research has proposed using an oscillating activation function to solve classification problems inspired by the human brain cortex. This paper explores the performance of one of the CNN architecture ALexNet on MNIST and CIFAR10 datasets using oscillatory acti…
Microwave Remote Sensing Lab Center of Studies in Resources Engineering（微波遥感实验室 资源工程研究中心）；Centre of Machine Intelligence and Data Science（智能与数据科学中心）；Department of Electrical Engineering（电气工程系）；School of Mathematics and Statistics（数学与统计学学院）；Center of Studies in Resources Engineering（资源工程研究中心）
Carnegie Mellon University（卡内基梅隆大学）；University of Alberta（阿尔伯塔大学）
We propose a Laplace approximation that creates a stochastic unit from any smooth monotonic activation function, using only Gaussian noise. This paper investigates the application of this stochastic approximation in training a family of Restricted Boltzmann Machines (RBM) that are closely linked to Bregman divergences. This family, that we call exponential family RBM (Exp-RBM), is a subset of the exponential family Harmoniums that expresses family members through a choice of smooth monotonic non-linearity for each neuron. Using contrastive divergence along with our Gaussian approximation, we s…
Activation Saturation and Floquet Spectrum Collapse in Neural ODEs
School of Pedagogical & Technological Education (ASPETE)（教育与技术教育学院（ASPETE））；Department of Mathematics, Chair of Dynamics, Control, Machine Learning and Numerics, Friedrich-Alexander-Universität Erlangen–Nürnberg (FAU)（弗里德里希-亚历山大-埃尔朗根-纽伦堡大学（FAU）数学系动力学、控制、机器学习与数值计算教席）
IIIT Delhi（德里英德拉普拉斯塔信息技术学院）；Shaheed Bhagat Singh College, University of Delhi（德里大学沙希德·巴格特·辛格学院）；University of Delhi（德里大学）
An activation function is a crucial component of a neural network that introduces non-linearity in the network. The state-of-the-art performance of a neural network depends also on the perfect choice of an activation function. We propose two novel non-monotonic smooth trainable activation functions, called ErfAct and Pserf. Experiments suggest that the proposed functions improve the network performance significantly compared to the widely used activations like ReLU, Swish, and Mish. Replacing ReLU by ErfAct and Pserf, we have 5.68% and 5.42% improvement for top-1 accuracy on Shufflenet V2 (2.0…
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, …
Indian Institute of Information Technology, Allahabad（印度理工学院信息技术学院（阿拉哈巴德））；Techno India University（泰克诺印度大学）；Indian Statistical Institute（印度统计研究所）
Neural networks have shown tremendous growth in recent years to solve numerous problems. Various types of neural networks have been introduced to deal with different types of problems. However, the main goal of any neural network is to transform the non-linearly separable input data into more linearly separable abstract features using a hierarchy of layers. These layers are combinations of linear and nonlinear functions. The most popular and common non-linearity layers are activation functions (AFs), such as Logistic Sigmoid, Tanh, ReLU, ELU, Swish and Mish. In this paper, a comprehensive over…
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…
Beijing University of Chemical Technology（北京化工大学）；Southwest Jiaotong University（西南交通大学）
Deep learning oriented named entity recognition (DNER) has gradually become the paradigm of knowledge discovery, which greatly promotes domain intelligence. However, the current activation function of DNER fails to treat gradient vanishing, no negative output or non-differentiable existence, which may impede knowledge exploration caused by the omission and incomplete representation of latent semantics. To break through the dilemma, we present a novel activation function termed KDAC. Detailly, KDAC is an aggregation function with multiple conversion modes. The backbone of the activation region …
IIIT Delhi（德里英迪拉·甘地信息技术学院）；Shaheed Bhagat Singh College（沙希德·巴格特·辛格学院）；University of Delhi（德里大学）
Activation functions play a pivotal role in the function learning using neural networks. The non-linearity in the learned function is achieved by repeated use of the activation function. Over the years, numerous activation functions have been proposed to improve accuracy in several tasks. Basic functions like ReLU, Sigmoid, Tanh, or Softplus have been favorite among the deep learning community because of their simplicity. In recent years, several novel activation functions arising from these basic functions have been proposed, which have improved accuracy in some challenging datasets. We propo…
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…
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 …
Dimensional Collapse in Transformer Attention Outputs: A Challenge for Sparse Dictionary Learning
Shanghai Innovation Institute, Shanghai, China（上海创新研究院）；OpenMOSS Team, School of Computer Science, Fudan University, Shanghai, China（开放MOSS团队，计算机科学学院，复旦大学）
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…
In this paper, we present MicroNet, which is an efficient convolutional neural network using extremely low computational cost (e.g. 6 MFLOPs on ImageNet classification). Such a low cost network is highly desired on edge devices, yet usually suffers from a significant performance degradation. We handle the extremely low FLOPs based upon two design principles: (a) avoiding the reduction of network width by lowering the node connectivity, and (b) compensating for the reduction of network depth by introducing more complex non-linearity per layer. Firstly, we propose Micro-Factorized convolution to…
SenseTime Research（商汤科技研究院）；The University of Hong Kong（香港大学）
Improving sparsity of deep neural networks (DNNs) is essential for network compression and has drawn much attention. In this work, we disclose a harmful sparsifying process called filter collapse, which is common in DNNs with batch normalization (BN) and rectified linear activation functions (e.g. ReLU, Leaky ReLU). It occurs even without explicit sparsity-inducing regularizations such as $L_1$. This phenomenon is caused by the normalization effect of BN, which induces a non-trainable region in the parameter space and reduces the network capacity as a result. This phenomenon becomes more promi…
The goal of a recommendation system is to predict the interest of a user in a given item by exploiting the existing set of ratings as well as certain user/item features. A standard approach to modeling this problem is Inductive Matrix Completion where the predicted rating is modeled as an inner product of the user and the item features projected onto a latent space. In order to learn the parameters effectively from a small number of observed ratings, the latent space is constrained to be low-dimensional which implies that the parameter matrix is constrained to be low-rank. However, such biline…
Neural Collapse under Gradient Flow on Shallow ReLU Networks for Orthogonally Separable Data
Institute of Natural Sciences & School of Mathematical Sciences, Shanghai Jiao Tong University（上海交通大学自然科学研究院与数学科学学院）；Computer Science and Engineering, Ohio State University（俄亥俄州立大学计算机科学与工程系）；Electrical and Systems Engineering & Department of Radiology, University of Pennsylvania（宾夕法尼亚大学电气与系统工程系及放射学系）
We propose quadratic residual networks (QRes) as a new type of parameter-efficient neural network architecture, by adding a quadratic residual term to the weighted sum of inputs before applying activation functions. With sufficiently high functional capacity (or expressive power), we show that it is especially powerful for solving forward and inverse physics problems involving partial differential equations (PDEs). Using tools from algebraic geometry, we theoretically demonstrate that, in contrast to plain neural networks, QRes shows better parameter efficiency in terms of network width and de…
Microsoft Research（微软研究院）；University at Buffalo（布法罗大学）
We present in this paper a systematic study on how to morph a well-trained neural network to a new one so that its network function can be completely preserved. We define this as \emph{network morphism} in this research. After morphing a parent network, the child network is expected to inherit the knowledge from its parent network and also has the potential to continue growing into a more powerful one with much shortened training time. The first requirement for this network morphism is its ability to handle diverse morphing types of networks, including changes of depth, width, kernel size, and…
We propose a new notion of `non-linearity' of a network layer with respect to an input batch that is based on its proximity to a linear system, which is reflected in the non-negative rank of the activation matrix. We measure this non-linearity by applying non-negative factorization to the activation matrix. Considering batches of similar samples, we find that high non-linearity in deep layers is indicative of memorization. Furthermore, by applying our approach layer-by-layer, we find that the mechanism for memorization consists of distinct phases. We perform experiments on fully-connected and …
