Kernel Methods and Multi-layer Perceptrons Learn Linear Models in High Dimensions
Halıcıoğlu Data Science Institute, UCSD（加州大学圣迭戈分校哈利西奥卢数据科学研究所）；Department of Electrical and Computer Engineering, NYU（纽约大学电气与计算机工程系）；Department of Electrical and Computer Engineering, UCLA（加州大学洛杉矶分校电气与计算机工程系）；Department of Statistics, UCLA（加州大学洛杉矶分校统计系）
Empirical observation of high dimensional phenomena, such as the double descent behaviour, has attracted a lot of interest in understanding classical techniques such as kernel methods, and their implications to explain generalization properties of neural networks. Many recent works analyze such models in a certain high-dimensional regime where the covariates are independent and the number of samples and the number of covariates grow at a fixed ratio (i.e. proportional asymptotics). In this work we show that for a large class of kernels, including the neural tangent kernel of fully connected networks, kernel methods can only perform as well as linear models in this regime. More surprisingly, when the data is generated by a kernel model where the relationship between input and the response could be very nonlinear, we show that linear models are in fact optimal, i.e. linear models achieve the minimum risk among all models, linear or nonlinear. These results suggest that more complex models for the data other than independent features are needed for high-dimensional analysis.
We propose Very Simple Classifier (VSC) a novel method designed to incorporate the concepts of subsampling and locality in the definition of features to be used as the input of a perceptron. The rationale is that locality theoretically guarantees a bound on the generalization error. Each feature in VSC is a max-margin classifier built on randomly-selected pairs of samples. The locality in VSC is achieved by multiplying the value of the feature by a confidence measure that can be characterized in terms of the Chebichev inequality. The output of the layer is then fed in a output layer of neurons…
We propose a new type of hidden layer for a multilayer perceptron, and demonstrate that it obtains the best reported performance for an MLP on the MNIST dataset.
In this paper, a simple, general method of adding auxiliary stochastic neurons to a multi-layer perceptron is proposed. It is shown that the proposed method is a generalization of recently successful methods of dropout (Hinton et al., 2012), explicit noise injection (Vincent et al., 2010; Bishop, 1995) and semantic hashing (Salakhutdinov & Hinton, 2009). Under the proposed framework, an extension of dropout which allows using separate dropping probabilities for different hidden neurons, or layers, is found to be available. The use of different dropping probabilities for hidden layers separatel…
University of Fortaleza (UNIFOR)（福塔莱萨大学（UNIFOR））；Federal University of ABC (UFABC)（ABC联邦大学（UFABC））
Perceptrons are neuronal devices capable of fully discriminating linearly separable classes. Although straightforward to implement and train, their applicability is usually hindered by non-trivial requirements imposed by real-world classification problems. Therefore, several approaches, such as kernel perceptrons, have been conceived to counteract such difficulties. In this paper, we investigate an enhanced perceptron model based on the notion of contrastive biclusters. From this perspective, a good discriminative bicluster comprises a subset of data instances belonging to one class that show …
It is well-known that the distribution over functions induced through a zero-mean iid prior distribution over the parameters of a multi-layer perceptron (MLP) converges to a Gaussian process (GP), under mild conditions. We extend this result firstly to independent priors with general zero or non-zero means, and secondly to a family of partially exchangeable priors which generalise iid priors. We discuss how the second prior arises naturally when considering an equivalence class of functions in an MLP and through training processes such as stochastic gradient descent. The model resulting from p…
University of Southern California（南加州大学）；Army Research Laboratory（陆军研究实验室）
A closed-form solution exists in two-class linear discriminant analysis (LDA), which discriminates two Gaussian-distributed classes in a multi-dimensional feature space. In this work, we interpret the multilayer perceptron (MLP) as a generalization of a two-class LDA system so that it can handle an input composed by multiple Gaussian modalities belonging to multiple classes. Besides input layer $l_{in}$ and output layer $l_{out}$, the MLP of interest consists of two intermediate layers, $l_1$ and $l_2$. We propose a feedforward design that has three stages: 1) from $l_{in}$ to $l_1$: half-spac…
Solving geometric tasks involving point clouds by using machine learning is a challenging problem. Standard feed-forward neural networks combine linear or, if the bias parameter is included, affine layers and activation functions. Their geometric modeling is limited, which motivated the prior work introducing the multilayer hypersphere perceptron (MLHP). Its constituent part, i.e., the hypersphere neuron, is obtained by applying a conformal embedding of Euclidean space. By virtue of Clifford algebra, it can be implemented as the Cartesian dot product of inputs and weights. If the embedding is …
McGill University（麦吉尔大学）；Mila - Quebec AI Institute（米拉 - 魁北克人工智能研究所）
Group invariant and equivariant Multilayer Perceptrons (MLP), also known as Equivariant Networks, have achieved remarkable success in learning on a variety of data structures, such as sequences, images, sets, and graphs. Using tools from group theory, this paper proves the universality of a broad class of equivariant MLPs with a single hidden layer. In particular, it is shown that having a hidden layer on which the group acts regularly is sufficient for universal equivariance (invariance). A corollary is unconditional universality of equivariant MLPs for Abelian groups, such as CNNs with a sin…
University of Tabriz（大不里士大学）；Urmia University（乌尔米耶大学）；Yazd University（亚兹德大学）；Obuda University（欧布达大学）；Duy Tan University（岘港大学（越南））；Ton Duc Thang University（东德大学）
Wind power as a renewable source of energy, has numerous economic, environmental and social benefits. In order to enhance and control renewable wind power, it is vital to utilize models that predict wind speed with high accuracy. Due to neglecting of requirement and significance of data preprocessing and disregarding the inadequacy of using a single predicting model, many traditional models have poor performance in wind speed prediction. In the current study, for predicting wind speed at target stations in the north of Iran, the combination of a multi-layer perceptron model (MLP) with the Whal…
Learning on 3D structures of large biomolecules is emerging as a distinct area in machine learning, but there has yet to emerge a unifying network architecture that simultaneously leverages the graph-structured and geometric aspects of the problem domain. To address this gap, we introduce geometric vector perceptrons, which extend standard dense layers to operate on collections of Euclidean vectors. Graph neural networks equipped with such layers are able to perform both geometric and relational reasoning on efficient and natural representations of macromolecular structure. We demonstrate our …
Autonomous construction of deep neural network (DNNs) is desired for data streams because it potentially offers two advantages: proper model's capacity and quick reaction to drift and shift. While the self-organizing mechanism of DNNs remains an open issue, this task is even more challenging to be developed for standard multi-layer DNNs than that using the different-depth structures, because the addition of a new layer results in information loss of previously trained knowledge. A Neural Network with Dynamically Evolved Capacity (NADINE) is proposed in this paper. NADINE features a fully open …
A Monte Carlo k-nearest neighbours (KNN) and a multi-resolution convolutional neural network (CNN) were developed to detect the presences of multiple gasses in near infrared (IR) spectrums. High Resolution Transmission database was used to synthesize the near IR spectrums. Monte Carlo KNN determined the optimal kernel sizes and the optimal number of channels. The multi-resolution CNN, composed of multiple different kernels, was created using the optimal kernel sizes and the optimal number of channels. The multi-resolution CNN outperforms the multilayer perceptron and the partial least squares.
New York University（纽约大学）；University of Amsterdam（阿姆斯特丹大学）
Symmetries and equivariance are fundamental to the generalization of neural networks on domains such as images, graphs, and point clouds. Existing work has primarily focused on a small number of groups, such as the translation, rotation, and permutation groups. In this work we provide a completely general algorithm for solving for the equivariant layers of matrix groups. In addition to recovering solutions from other works as special cases, we construct multilayer perceptrons equivariant to multiple groups that have never been tackled before, including $\mathrm{O}(1,3)$, $\mathrm{O}(5)$, $\mat…
Tel Aviv University（特拉维夫大学）；New York University（纽约大学）
A major factor in the success of deep neural networks is the use of sophisticated architectures rather than the classical multilayer perceptron (MLP). Residual networks (ResNets) stand out among these powerful modern architectures. Previous works focused on the optimization advantages of deep ResNets over deep MLPs. In this paper, we show another distinction between the two models, namely, a tendency of ResNets to promote smoother interpolations than MLPs. We analyze this phenomenon via the neural tangent kernel (NTK) approach. First, we compute the NTK for a considered ResNet model and prove …
Seoul National University（首尔国立大学）；Hongik University（弘益大学）
The tremendous recent progress in analyzing the training dynamics of overparameterized neural networks has primarily focused on wide networks and therefore does not sufficiently address the role of depth in deep learning. In this work, we present the first trainability guarantee of infinitely deep but narrow neural networks. We study the infinite-depth limit of a multilayer perceptron (MLP) with a specific initialization and establish a trainability guarantee using the NTK theory. We then extend the analysis to an infinitely deep convolutional neural network (CNN) and perform brief experiments…
CSIRO（澳大利亚联邦科学与工业研究组织）；The University of Queensland（昆士兰大学）；University of Cambridge（剑桥大学）；International Computer Science Institute（国际计算机科学研究所）
Analysing and computing with Gaussian processes arising from infinitely wide neural networks has recently seen a resurgence in popularity. Despite this, many explicit covariance functions of networks with activation functions used in modern networks remain unknown. Furthermore, while the kernels of deep networks can be computed iteratively, theoretical understanding of deep kernels is lacking, particularly with respect to fixed-point dynamics. Firstly, we derive the covariance functions of multi-layer perceptrons (MLPs) with exponential linear units (ELU) and Gaussian error linear units (GELU)…
Mitsubishi Electric Research Laboratories (MERL)（三菱电机研究实验室（MERL））；Cambridge, MA 02139-1955（剑桥，马萨诸塞州 02139-1955（注意：此部分为地址，不应作为机构提取））
In this paper we propose solving localized multiple kernel learning (LMKL) using LMKL-Net, a feedforward deep neural network. In contrast to previous works, as a learning principle we propose {\em parameterizing} both the gating function for learning kernel combination weights and the multiclass classifier in LMKL using an attentional network (AN) and a multilayer perceptron (MLP), respectively. In this way we can learn the (nonlinear) decision function in LMKL (approximately) by sequential applications of AN and MLP. Empirically on benchmark datasets we demonstrate that overall LMKL-Net can n…
Prominently used in support vector machines and logistic regressions, kernel functions (kernels) can implicitly map data points into high dimensional spaces and make it easier to learn complex decision boundaries. In this work, by replacing the inner product function in the softmax layer, we explore the use of kernels for contextual word classification. In order to compare the individual kernels, experiments are conducted on standard language modeling and machine translation tasks. We observe a wide range of performances across different kernel settings. Extending the results, we look at the g…
