Deep neural networks (DNNs) achieve impressive results for complicated tasks like object detection on images and speech recognition. Motivated by this practical success, there is now a strong interest in showing good theoretical properties of DNNs. To describe for which tasks DNNs perform well and when they fail, it is a key challenge to understand their performance. The aim of this paper is to contribute to the current statistical theory of DNNs. We apply DNNs on high dimensional data and we show that the least squares regression estimates using DNNs are able to achieve dimensionality reduction in case that the regression function has locally low dimensionality. Consequently, the rate of convergence of the estimate does not depend on its input dimension $d$, but on its local dimension $d^*$ and the DNNs are able to circumvent the curse of dimensionality in case that $d^*$ is much smaller than $d$. In our simulation study we provide numerical experiments to support our theoretical result and we compare our estimate with other conventional nonparametric regression estimates. The performance of our estimates is also validated in experiments with real data.
Breaking the curse of dimensionality in structured density estimation
Berlin Institute for the Foundations of Learning and Data (BIFOLD)（柏林学习与数据基础研究所（BIFOLD））；Technische Universität Berlin（柏林工业大学）；Nanyang Technological University（南洋理工大学）
University of California, San Diego（加利福尼亚大学圣迭戈分校）；Halıcıǧlu Data Science Institute, University of California San Diego（加利福尼亚大学圣迭戈分校Halıcıǧlu数据科学研究所）；Simula Research Laboratory（Simula研究实验室）
We study the approximation of two-layer compositions $f(x) = g(ϕ(x))$ via deep networks with ReLU activation, where $ϕ$ is a geometrically intuitive, dimensionality reducing feature map. We focus on two intuitive and practically relevant choices for $ϕ$: the projection onto a low-dimensional embedded submanifold and a distance to a collection of low-dimensional sets. We achieve near optimal approximation rates, which depend only on the complexity of the dimensionality reducing map $ϕ$ rather than the ambient dimension. Since $ϕ$ encapsulates all nonlinear features that are material to the func…
Regression aims at estimating the conditional mean of output given input. However, regression is not informative enough if the conditional density is multimodal, heteroscedastic, and asymmetric. In such a case, estimating the conditional density itself is preferable, but conditional density estimation (CDE) is challenging in high-dimensional space. A naive approach to coping with high-dimensionality is to first perform dimensionality reduction (DR) and then execute CDE. However, such a two-step process does not perform well in practice because the error incurred in the first DR step can be mag…
Neural networks appear to have mysterious generalization properties when using parameter counting as a proxy for complexity. Indeed, neural networks often have many more parameters than there are data points, yet still provide good generalization performance. Moreover, when we measure generalization as a function of parameters, we see double descent behaviour, where the test error decreases, increases, and then again decreases. We show that many of these properties become understandable when viewed through the lens of effective dimensionality, which measures the dimensionality of the parameter…
Northeastern University（东北大学）；Graz University of Technology（格拉茨技术大学）；Silicon Austria Labs（硅谷奥地利实验室）
Feature ranking and selection is a widely used approach in various applications of supervised dimensionality reduction in discriminative machine learning. Nevertheless there exists significant evidence on feature ranking and selection algorithms based on any criterion leading to potentially sub-optimal solutions for class separability. In that regard, we introduce emerging information theoretic feature transformation protocols as an end-to-end neural network training approach. We present a dimensionality reduction network (MMINet) training procedure based on the stochastic estimate of the mutu…
In recent years, manifold learning has become increasingly popular as a tool for performing non-linear dimensionality reduction. This has led to the development of numerous algorithms of varying degrees of complexity that aim to recover man ifold geometry using either local or global features of the data. Building on the Laplacian Eigenmap and Diffusionmaps framework, we propose a new paradigm that offers a guarantee, under reasonable assumptions, that any manifo ld learning algorithm will preserve the geometry of a data set. Our approach is based on augmenting the output of embedding algorith…
Machine learning algorithms using deep architectures have been able to implement increasingly powerful and successful models. However, they also become increasingly more complex, more difficult to comprehend and easier to fool. So far, most methods in the literature investigate the decision of the model for a single given input datum. In this paper, we propose to visualize a part of the decision function of a deep neural network together with a part of the data set in two dimensions with discriminative dimensionality reduction. This enables us to inspect how different properties of the data ar…
University of Waterloo（滑铁卢大学）；Vector Institute（向量研究所）
We consider the problem of sufficient dimensionality reduction (SDR), where the high-dimensional observation is transformed to a low-dimensional sub-space in which the information of the observations regarding the label variable is preserved. We propose DVSDR, a deep variational approach for sufficient dimensionality reduction. The deep structure in our model has a bottleneck that represent the low-dimensional embedding of the data. We explain the SDR problem using graphical models and use the framework of variational autoencoders to maximize the lower bound of the log-likelihood of the joint …
University of Wisconsin–Madison（威斯康星大学麦迪逊分校）；École Polytechnique Fédérale de Lausanne（洛桑联邦理工学院）
We study the problem of estimating an unknown function from noisy data using shallow ReLU neural networks. The estimators we study minimize the sum of squared data-fitting errors plus a regularization term proportional to the squared Euclidean norm of the network weights. This minimization corresponds to the common approach of training a neural network with weight decay. We quantify the performance (mean-squared error) of these neural network estimators when the data-generating function belongs to the second-order Radon-domain bounded variation space. This space of functions was recently propo…
Sun Yat-sen University（中山大学）；RIKEN Center for Brain Science（理化学研究所脑科学中心）
Deep neural networks are widely used in various domains. However, the nature of computations at each layer of the deep networks is far from being well understood. Increasing the interpretability of deep neural networks is thus important. Here, we construct a mean-field framework to understand how compact representations are developed across layers, not only in deterministic deep networks with random weights but also in generative deep networks where an unsupervised learning is carried out. Our theory shows that the deep computation implements a dimensionality reduction while maintaining a fini…
The University of Tokyo（东京大学）；RIKEN Center for Advanced Intelligence Project（理化学研究所先进智能项目中心）；Kyushu Institute of Technology（九州工业大学）
Deep learning has exhibited superior performance for various tasks, especially for high-dimensional datasets, such as images. To understand this property, we investigate the approximation and estimation ability of deep learning on anisotropic Besov spaces. The anisotropic Besov space is characterized by direction-dependent smoothness and includes several function classes that have been investigated thus far. We demonstrate that the approximation error and estimation error of deep learning only depend on the average value of the smoothness parameters in all directions. Consequently, the curse o…
Manifold learning-based encoders have been playing important roles in nonlinear dimensionality reduction (NLDR) for data exploration. However, existing methods can often fail to preserve geometric, topological and/or distributional structures of data. In this paper, we propose a deep manifold learning framework, called deep manifold transformation (DMT) for unsupervised NLDR and embedding learning. DMT enhances deep neural networks by using cross-layer local geometry-preserving (LGP) constraints. The LGP constraints constitute the loss for deep manifold learning and serve as geometric regulari…
New Jersey Institute of Technology（新泽西理工学院）；Bremen University of Applied Sciences（不来梅应用技术大学）；Princeton University（普林斯顿大学）；Technion - Israel Institute of Technology（以色列理工学院）
A common way of characterizing minimax estimators in point estimation is by moving the problem into the Bayesian estimation domain and finding a least favorable prior distribution. The Bayesian estimator induced by a least favorable prior, under mild conditions, is then known to be minimax. However, finding least favorable distributions can be challenging due to inherent optimization over the space of probability distributions, which is infinite-dimensional. This paper develops a dimensionality reduction method that allows us to move the optimization to a finite-dimensional setting with an exp…
The University of Tokyo（东京大学）；RIKEN（理化学研究所）；Japan Digital Design（日本数字设计公司）
Deep learning has shown high performances in various types of tasks from visual recognition to natural language processing, which indicates superior flexibility and adaptivity of deep learning. To understand this phenomenon theoretically, we develop a new approximation and estimation error analysis of deep learning with the ReLU activation for functions in a Besov space and its variant with mixed smoothness. The Besov space is a considerably general function space including the Holder space and Sobolev space, and especially can capture spatial inhomogeneity of smoothness. Through the analysis …
University of Washington（华盛顿大学）；Harvard University（哈佛大学）；Université de Montréal（蒙特利尔大学）
Datasets such as images, text, or movies are embedded in high-dimensional spaces. However, in important cases such as images of objects, the statistical structure in the data constrains samples to a manifold of dramatically lower dimensionality. Learning to identify and extract task-relevant variables from this embedded manifold is crucial when dealing with high-dimensional problems. We find that neural networks are often very effective at solving this task and investigate why. To this end, we apply state-of-the-art techniques for intrinsic dimensionality estimation to show that neural network…
University of Virginia（弗吉尼亚大学）；University of Minnesota（明尼苏达大学）；Amazon（亚马逊）
Learning generative probabilistic models is a core problem in machine learning, which presents significant challenges due to the curse of dimensionality. This paper proposes a joint dimensionality reduction and non-parametric density estimation framework, using a novel estimator that can explicitly capture the underlying distribution of appropriate reduced-dimension representations of the input data. The idea is to jointly design a nonlinear dimensionality reducing auto-encoder to model the training data in terms of a parsimonious set of latent random variables, and learn a canonical low-rank …
Rutgers University（罗格斯大学）；The University of Tokyo（东京大学）；Komaba Institute for Science, The University of Tokyo（东京大学驹场科学研究所）；Center for Advanced Intelligence Project, RIKEN（理化学研究所革新智能综合研究中心）
In this study, we prove that an intrinsic low dimensionality of covariates is the main factor that determines the performance of deep neural networks (DNNs). DNNs generally provide outstanding empirical performance. Hence, numerous studies have actively investigated the theoretical properties of DNNs to understand their underlying mechanisms. In particular, the behavior of DNNs in terms of high-dimensional data is one of the most critical questions. However, this issue has not been sufficiently investigated from the aspect of covariates, although high-dimensional data have practically low intr…
Unsupervised deep learning is one of the most powerful representation learning techniques. Restricted Boltzman machine, sparse coding, regularized auto-encoders, and convolutional neural networks are pioneering building blocks of deep learning. In this paper, we propose a new building block -- distributed random models. The proposed method is a special full implementation of the product of experts: (i) each expert owns multiple hidden units and different experts have different numbers of hidden units; (ii) the model of each expert is a k-center clustering, whose k-centers are only uniformly sa…
This paper addresses the following questions pertaining to the intrinsic dimensionality of any given image representation: (i) estimate its intrinsic dimensionality, (ii) develop a deep neural network based non-linear mapping, dubbed DeepMDS, that transforms the ambient representation to the minimal intrinsic space, and (iii) validate the veracity of the mapping through image matching in the intrinsic space. Experiments on benchmark image datasets (LFW, IJB-C and ImageNet-100) reveal that the intrinsic dimensionality of deep neural network representations is significantly lower than the dimens…
We consider the problem of approximating the set of eigenvalues of the covariance matrix of a multivariate distribution (equivalently, the problem of approximating the "population spectrum"), given access to samples drawn from the distribution. The eigenvalues of the covariance of a distribution contain basic information about the distribution, including the presence or lack of structure in the distribution, the effective dimensionality of the distribution, and the applicability of higher-level machine learning and multivariate statistical tools. We consider this fundamental recovery problem i…
Technische Universität Berlin（柏林工业大学）；Machine Learning Group（机器学习组）
While nonparametric density estimators often perform well on low dimensional data, their performance can suffer when applied to higher dimensional data, owing presumably to the curse of dimensionality. One technique for avoiding this is to assume no dependence between features and that the data are sampled from a separable density. This allows one to estimate each marginal distribution independently thereby avoiding the slow rates associated with estimating the full joint density. This is a strategy employed in naive Bayes models and is analogous to estimating a rank-one tensor. In this paper …
Mitsubishi Electric Corporation（三菱电机株式会社）；Mitsubishi Electric Research Laboratory（三菱电机研究实验室）
Deep reinforcement learning (RL) algorithms have recently achieved remarkable successes in various sequential decision making tasks, leveraging advances in methods for training large deep networks. However, these methods usually require large amounts of training data, which is often a big problem for real-world applications. One natural question to ask is whether learning good representations for states and using larger networks helps in learning better policies. In this paper, we try to study if increasing input dimensionality helps improve performance and sample efficiency of model-free deep…
University of Vienna（维也纳大学）；ETH Zürich（苏黎世联邦理工学院）；University of Münster（明斯特大学）
The development of new classification and regression algorithms based on empirical risk minimization (ERM) over deep neural network hypothesis classes, coined deep learning, revolutionized the area of artificial intelligence, machine learning, and data analysis. In particular, these methods have been applied to the numerical solution of high-dimensional partial differential equations with great success. Recent simulations indicate that deep learning-based algorithms are capable of overcoming the curse of dimensionality for the numerical solution of Kolmogorov equations, which are widely used i…
In the last decades the estimation of the intrinsic dimensionality of a dataset has gained considerable importance. Despite the great deal of research work devoted to this task, most of the proposed solutions prove to be unreliable when the intrinsic dimensionality of the input dataset is high and the manifold where the points lie is nonlinearly embedded in a higher dimensional space. In this paper we propose a novel robust intrinsic dimensionality estimator that exploits the twofold complementary information conveyed both by the normalized nearest neighbor distances and by the angles computed…
Universitá di Pisa（比萨大学）；Universitá degli Studi di Cagliari（卡利亚里大学）
Deep neural networks are vulnerable to adversarial examples, i.e., carefully-perturbed inputs aimed to mislead classification. This work proposes a detection method based on combining non-linear dimensionality reduction and density estimation techniques. Our empirical findings show that the proposed approach is able to effectively detect adversarial examples crafted by non-adaptive attackers, i.e., not specifically tuned to bypass the detection method. Given our promising results, we plan to extend our analysis to adaptive attackers in future work.
Correct risk estimation of policyholders is of great significance to auto insurance companies. While the current tools used in this field have been proven in practice to be quite efficient and beneficial, we argue that there is still a lot of room for development and improvement in the auto insurance risk estimation process. To this end, we develop a framework based on a combination of a neural network together with a dimensionality reduction technique t-SNE (t-distributed stochastic neighbour embedding). This enables us to visually represent the complex structure of the risk as a two-dimensio…
University of Warsaw（华沙大学）；Polish National Institute for Machine Learning（波兰国家机器学习研究所）；Applica；Jagiellonian University（雅盖隆大学）；University of Oxford（牛津大学）
Most of the existing methods for estimating the local intrinsic dimension of a data distribution do not scale well to high-dimensional data. Many of them rely on a non-parametric nearest neighbors approach which suffers from the curse of dimensionality. We attempt to address that challenge by proposing a novel approach to the problem: Local Intrinsic Dimension estimation using approximate Likelihood (LIDL). Our method relies on an arbitrary density estimation method as its subroutine and hence tries to sidestep the dimensionality challenge by making use of the recent progress in parametric neu…
Hashing is a basic tool for dimensionality reduction employed in several aspects of machine learning. However, the perfomance analysis is often carried out under the abstract assumption that a truly random unit cost hash function is used, without concern for which concrete hash function is employed. The concrete hash function may work fine on sufficiently random input. The question is if it can be trusted in the real world when faced with more structured input. In this paper we focus on two prominent applications of hashing, namely similarity estimation with the one permutation hashing (OPH) s…
In this paper, we demonstrate a computationally efficient new approach based on deep learning (DL) techniques for analysis, design, and optimization of electromagnetic (EM) nanostructures. We use the strong correlation among features of a generic EM problem to considerably reduce the dimensionality of the problem and thus, the computational complexity, without imposing considerable errors. By employing the dimensionality reduction concept using the more recently demonstrated autoencoder technique, we redefine the conventional many-to-one design problem in EM nanostructures into a one-to-one pr…
Unsupervised learning of time series data, also known as temporal clustering, is a challenging problem in machine learning. Here we propose a novel algorithm, Deep Temporal Clustering (DTC), to naturally integrate dimensionality reduction and temporal clustering into a single end-to-end learning framework, fully unsupervised. The algorithm utilizes an autoencoder for temporal dimensionality reduction and a novel temporal clustering layer for cluster assignment. Then it jointly optimizes the clustering objective and the dimensionality reduction objec tive. Based on requirement and application, …
In order to identify important variables that are involved in making optimal treatment decision, Lu et al. (2013) proposed a penalized least squared regression framework for a fixed number of predictors, which is robust against the misspecification of the conditional mean model. Two problems arise: (i) in a world of explosively big data, effective methods are needed to handle ultra-high dimensional data set, for example, with the dimension of predictors is of the non-polynomial (NP) order of the sample size; (ii) both the propensity score and conditional mean models need to be estimated from d…
Coventry University（考文垂大学）；Malayer University（马拉耶尔大学）；Cranfield University（克兰菲尔德大学）；The University of Warwick（华威大学）
A computational method based on the non-linear Gaussian process (GP), known as deep Gaussian processes (deep GPs) for uncertainty quantification & propagation in modelling of flow through heterogeneous porous media is presented. The method is also used for reducing dimensionality of model output and consequently emulating highly complex relationship between hydrogeological properties and reduced order fluid velocity field in a tractable manner. Deep GPs are multi-layer hierarchical generalisations of GPs with multiple, infinitely wide hidden layers that are very efficient models for deep learn…
Training deep learning models that generalize well to live deployment is a challenging problem in the financial markets. The challenge arises because of high dimensionality, limited observations, changing data distributions, and a low signal-to-noise ratio. High dimensionality can be dealt with using robust feature selection or dimensionality reduction, but limited observations often result in a model that overfits due to the large parameter space of most deep neural networks. We propose a generative model for financial time series, which allows us to train deep learning models on millions of …
George Washington University（乔治华盛顿大学）；National Institute of Standards and Technology（美国国家标准与技术研究院）
The movement of large quantities of data during the training of a Deep Neural Network presents immense challenges for machine learning workloads. To minimize this overhead, especially on the movement and calculation of gradient information, we introduce streaming batch principal component analysis as an update algorithm. Streaming batch principal component analysis uses stochastic power iterations to generate a stochastic k-rank approximation of the network gradient. We demonstrate that the low rank updates produced by streaming batch principal component analysis can effectively train convolut…
Despite the fact that they do not consider the temporal nature of data, classic dimensionality reduction techniques, such as PCA, are widely applied to time series data. In this paper, we introduce a factor decomposition specific for time series that builds upon the Bayesian multivariate autoregressive model and hence evades the assumption that data points are mutually independent. The key is to find a low-rank estimation of the autoregressive matrices. As in the probabilistic version of other factor models, this induces a latent low-dimensional representation of the original data. We discuss …
Dimensionality reduction is a topic of recent interest. In this paper, we present the classification constrained dimensionality reduction (CCDR) algorithm to account for label information. The algorithm can account for multiple classes as well as the semi-supervised setting. We present an out-of-sample expressions for both labeled and unlabeled data. For unlabeled data, we introduce a method of embedding a new point as preprocessing to a classifier. For labeled data, we introduce a method that improves the embedding during the training phase using the out-of-sample extension. We investigate cl…
Multivariate Density Estimation via Variance-Reduced Sketching
Committee on Computational and Applied Mathematics, University of Chicago（芝加哥大学计算与应用数学委员会）；Department of Statistics, University of Chicago（芝加哥大学统计学系）；Department of Applied and Computational Mathematics and Statistics, University of Notre Dame（南达科他大学应用与计算数学与统计学系）
University of Oxford（牛津大学）；The Alan Turing Institute（艾伦·图灵研究所）
We investigate the effect of the dimensionality of the representations learned in Deep Neural Networks (DNNs) on their robustness to input perturbations, both adversarial and random. To achieve low dimensionality of learned representations, we propose an easy-to-use, end-to-end trainable, low-rank regularizer (LR) that can be applied to any intermediate layer representation of a DNN. This regularizer forces the feature representations to (mostly) lie in a low-dimensional linear subspace. We perform a wide range of experiments that demonstrate that the LR indeed induces low rank on the represen…
Machine Learning models are vulnerable to adversarial attacks that rely on perturbing the input data. This work proposes a novel strategy using Autoencoder Deep Neural Networks to defend a machine learning model against two gradient-based attacks: The Fast Gradient Sign attack and Fast Gradient attack. First we use an autoencoder to denoise the test data, which is trained with both clean and corrupted data. Then, we reduce the dimension of the denoised data using the hidden layer representation of another autoencoder. We perform this experiment for multiple values of the bound of adversarial p…
Duke University（杜克大学）；Oak Ridge National Laboratory（橡树岭国家实验室）
Disentangled learning representations have promising utility in many applications, but they currently suffer from serious reliability issues. We present Gaussian Channel Autoencoder (GCAE), a method which achieves reliable disentanglement via flexible density estimation of the latent space. GCAE avoids the curse of dimensionality of density estimation by disentangling subsets of its latent space with the Dual Total Correlation (DTC) metric, thereby representing its high-dimensional latent joint distribution as a collection of many low-dimensional conditional distributions. In our experiments, …
This paper considers improved forecasting in possibly nonlinear dynamic settings, with high-dimension predictors ("big data" environments). To overcome the curse of dimensionality and manage data and model complexity, we examine shrinkage estimation of a back-propagation algorithm of a deep neural net with skip-layer connections. We expressly include both linear and nonlinear components. This is a high-dimensional learning approach including both sparsity L1 and smoothness L2 penalties, allowing high-dimensionality and nonlinearity to be accommodated in one step. This approach selects signific…
Wigner Research Centre for Physics（维格纳物理研究中心）；Semmelweis University（塞梅尔维斯大学）；National Institute of Clinical Neurosciences（国家临床神经科学研究所）；Péter Pázmány Catholic University（帕兹马尼·彼得天主教大学）；Budapest University of Technology and Economics（布达佩斯技术与经济大学）；University of Pannonia（潘诺尼亚大学）
Data dimensionality informs us about data complexity and sets limit on the structure of successful signal processing pipelines. In this work we revisit and improve the manifold-adaptive Farahmand-Szepesvári-Audibert (FSA) dimension estimator, making it one of the best nearest neighbor-based dimension estimators available. We compute the probability density function of local FSA estimates, if the local manifold density is uniform. Based on the probability density function, we propose to use the median of local estimates as a basic global measure of intrinsic dimensionality, and we demonstrate t…
Approximate Message Passing (AMP) has been shown to be a superior method for inference problems, such as the recovery of signals from sets of noisy, lower-dimensionality measurements, both in terms of reconstruction accuracy and in computational efficiency. However, AMP suffers from serious convergence issues in contexts that do not exactly match its assumptions. We propose a new approach to stabilizing AMP in these contexts by applying AMP updates to individual coefficients rather than in parallel. Our results show that this change to the AMP iteration can provide theoretically expected, but …
