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 {\it 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 of dimensionality can be avoided if the smoothness of the target function is highly anisotropic. Unlike existing studies, our analysis does not require a low-dimensional structure of the input data. We also investigate the minimax optimality of deep learning and compare its performance with that of the kernel method (more generally, linear estimators). The results show that deep learning has better dependence on the input dimensionality if the target function possesses anisotropic smoothness, and it achieves an adaptive rate for functions with spatially inhomogeneous smoothness.

Manipulating data, such as weighting data examples or augmenting with new instances, has been increasingly used to improve model training. Previous work has studied various rule- or learning-based approaches designed for specific types of data manipulation. In this work, we propose a new method that supports learning different manipulation schemes with the same gradient-based algorithm. Our approach builds upon a recent connection of supervised learning and reinforcement learning (RL), and adapts an off-the-shelf reward learning algorithm from RL for joint data manipulation learning and model training. Different parameterization of the "data reward" function instantiates different manipulation schemes. We showcase data augmentation that learns a text transformation network, and data weighting that dynamically adapts the data sample importance. Experiments show the resulting algorithms significantly improve the image and text classification performance in low data regime and class-imbalance problems.

The performance of gradient-based optimization strategies depends heavily on the initial weights of the parametric model. Recent works show that there exist weight initializations from which optimization procedures can find the task-specific parameters faster than from uniformly random initializations, and that such a weight initialization can be learned by optimizing a specific model architecture across similar tasks via MAML (Model-Agnostic Meta-Learning). Current methods are limited to populations of classification tasks that share the same number of classes due to the static model architectures used during meta-learning. In this paper, we present HIDRA, a meta-learning approach that enables training and evaluating across tasks with any number of target variables. We show that Model-Agnostic Meta-Learning trains a distribution for all the neurons in the output layer and a specific weight initialization for the ones in the hidden layers. HIDRA explores this by learning one master neuron which is used to initialize any number of output neurons for a new task. Extensive experiments on the Miniimagenet and Omniglot data sets demonstrate that HIDRA improves over standard approaches while generalizing to tasks with any number of target variables. Moreover, our approach is shown to robustify low-capacity models in learning across complex tasks with a high number of classes for which regular MAML fails to learn any feasible initialization.

Deep neural networks (DNN) have achieved tremendous success in many areas, including image processing, natural language processing, video, and audio synthesis. They have also been used for a long time as a general tool for solving regression tasks, i.e., an approximation of a given function from its samples. Neural networks are known to be a universal approximator for continuous functions [6, 3]. Recently, several approximation rate results have been established: it has been shown that a certain class of deep neural networks with ReLU [11] activation functions provide guaranteed convergence rates for certain function classes [16, 5, 9]. Recent paper [12] provides expressive power results for general piecewise analytic functions with point singularities.

Discovering the underlying physical behavior of complex systems is a crucial, but less well-understood topic in many engineering disciplines. This study proposes a finite-difference inspired convolutional neural network framework to learn hidden partial differential equations from given data and iteratively estimate future dynamical behavior. The methodology designs the filter sizes such that they mimic the finite difference between the neighboring points. By learning the governing equation, the network predicts the future evolution of the solution by using only a few trainable parameters. In this paper, we provide numerical results to compare the efficiency of the second-order Trust-Region Conjugate Gradient (TRCG) method with the first-order ADAM optimizer.

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 gradient properties, investigate various mixture strategies and examine the disambiguation abilities.

Traditionally, the vision community has devised algorithms to estimate the distance between an original image and images that have been subject to perturbations. Inspiration was usually taken from the human visual perceptual system and how the system processes different perturbations in order to replicate to what extent it determines our ability to judge image quality. While recent works have presented deep neural networks trained to predict human perceptual quality, very few borrow any intuitions from the human visual system. To address this, we present PerceptNet, a convolutional neural network where the architecture has been chosen to reflect the structure and various stages in the human visual system. We evaluate PerceptNet on various traditional perception datasets and note strong performance on a number of them as compared with traditional image quality metrics. We also show that including a nonlinearity inspired by the human visual system in classical deep neural networks architectures can increase their ability to judge perceptual similarity.

We propose Generative Well-intentioned Networks (GWINs), a novel framework for increasing the accuracy of certainty-based, closed-world classifiers. A conditional generative network recovers the distribution of observations that the classifier labels correctly with high certainty. We introduce a reject option to the classifier during inference, allowing the classifier to reject an observation instance rather than predict an uncertain label. These rejected observations are translated by the generative network to high-certainty representations, which are then relabeled by the classifier. This architecture allows for any certainty-based classifier or rejection function and is not limited to multilayer perceptrons. The capability of this framework is assessed using benchmark classification datasets and shows that GWINs significantly improve the accuracy of uncertain observations.

We study the problem of object detection over scanned images of scientific documents. We consider images that contain objects of varying aspect ratios and sizes and range from coarse elements such as tables and figures to fine elements such as equations and section headers. We find that current object detectors fail to produce properly localized region proposals over such page objects. We revisit the original R-CNN model and present a method for generating fine-grained proposals over document elements. We also present a region embedding model that uses the convolutional maps of a proposal's neighbors as context to produce an embedding for each proposal. This region embedding is able to capture the semantic relationships between a target region and its surrounding context. Our end-to-end model produces an embedding for each proposal, then classifies each proposal by using a multi-head attention model that attends to the most important neighbors of a proposal. To evaluate our model, we collect and annotate a dataset of publications from heterogeneous journals. We show that our model, referred to as Attentive-RCNN, yields a 17% mAP improvement compared to standard object detection models.

Stochastic-sampling-based Generative Neural Networks, such as Restricted Boltzmann Machines and Generative Adversarial Networks, are now used for applications such as denoising, image occlusion removal, pattern completion, and motion synthesis. In scenarios which involve performing such inference tasks with these models, it is critical to determine metrics that allow for model selection and/or maintenance of requisite generative performance under pre-specified implementation constraints. In this paper, we propose a new metric for evaluating generative model performance based on $p$-values derived from the combined use of Maximum Mean Discrepancy (MMD) and permutation-based (PT-based) resampling, which we refer to as PT-MMD. We demonstrate the effectiveness of this metric for two cases: (1) Selection of bitwidth and activation function complexity to achieve minimum power-at-performance for Restricted Boltzmann Machines; (2) Quantitative comparison of images generated by two types of Generative Adversarial Networks (PGAN and WGAN) to facilitate model selection in order to maximize the fidelity of generated images. For these applications, our results are shown using Euclidean and Haar-based kernels for the PT-MMD two sample hypothesis test. This demonstrates the critical role of distance functions in comparing generated images against their corresponding ground truth counterparts as what would be perceived by human users.