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Deep learning has shown successful application in visual recognition and certain artificial intelligence tasks. Deep learning is also considered as a powerful tool with high flexibility to approximate functions. In the present work, functions with desired properties are devised to approximate the solutions of PDEs. Our approach is based on a posteriori error estimation in which the adjoint problem is solved for the error localization to formulate an error estimator within the framework of neural network. An efficient and easy to implement algorithm is developed to obtain a posteriori error estimate for multiple goal functionals by employing the dual-weighted residual approach, which is followed by the computation of both primal and adjoint solutions using the neural network. The present study shows that such a data-driven model based learning has superior approximation of quantities of interest even with relatively less training data. The novel algorithmic developments are substantiated with numerical test examples. The advantages of using deep neural network over the shallow neural network are demonstrated and the convergence enhancing techniques are also presented

In this study, we develop an asymptotic theory of nonparametric regression for a locally stationary functional time series. First, we introduce the notion of a locally stationary functional time series (LSFTS) that takes values in a semi-metric space. Then, we propose a nonparametric model for LSFTS with a regression function that changes smoothly over time. We establish the uniform convergence rates of a class of kernel estimators, the Nadaraya-Watson (NW) estimator of the regression function, and a central limit theorem of the NW estimator.

Various approaches have been proposed for out-of-distribution (OOD) detection by augmenting models, input examples, training sets, and optimization objectives. Deviating from existing work, we have a simple hypothesis that standard off-the-shelf models may already contain sufficient information about the training set distribution which can be leveraged for reliable OOD detection. Our empirical study on validating this hypothesis, which measures the model activation's mean for OOD and in-distribution (ID) mini-batches, surprisingly finds that activation means of OOD mini-batches consistently deviate more from those of the training data. In addition, training data's activation means can be computed offline efficiently or retrieved from batch normalization layers as a `free lunch'. Based upon this observation, we propose a novel metric called Neural Mean Discrepancy (NMD), which compares neural means of the input examples and training data. Leveraging the simplicity of NMD, we propose an efficient OOD detector that computes neural means by a standard forward pass followed by a lightweight classifier. Extensive experiments show that NMD outperforms state-of-the-art OOD approaches across multiple datasets and model architectures in terms of both detection accuracy and computational cost.

Blind deconvolution is an ill-posed problem arising in various fields ranging from microscopy to astronomy. The ill-posed nature of the problem requires adequate priors to arrive to a desirable solution. Recently, it has been shown that deep learning architectures can serve as an image generation prior during unsupervised blind deconvolution optimization, however often exhibiting a performance fluctuation even on a single image. We propose to use Wiener-deconvolution to guide the image generator during optimization by providing it a sharpened version of the blurry image using an auxiliary kernel estimate starting from a Gaussian. We observe that the high-frequency artifacts of deconvolution are reproduced with a delay compared to low-frequency features. In addition, the image generator reproduces low-frequency features of the deconvolved image faster than that of a blurry image. We embed the computational process in a constrained optimization framework and show that the proposed method yields higher stability and performance across multiple datasets. In addition, we provide the code.

Heatmap-based methods dominate in the field of human pose estimation by modelling the output distribution through likelihood heatmaps. In contrast, regression-based methods are more efficient but suffer from inferior performance. In this work, we explore maximum likelihood estimation (MLE) to develop an efficient and effective regression-based methods. From the perspective of MLE, adopting different regression losses is making different assumptions about the output density function. A density function closer to the true distribution leads to a better regression performance. In light of this, we propose a novel regression paradigm with Residual Log-likelihood Estimation (RLE) to capture the underlying output distribution. Concretely, RLE learns the change of the distribution instead of the unreferenced underlying distribution to facilitate the training process. With the proposed reparameterization design, our method is compatible with off-the-shelf flow models. The proposed method is effective, efficient and flexible. We show its potential in various human pose estimation tasks with comprehensive experiments. Compared to the conventional regression paradigm, regression with RLE bring 12.4 mAP improvement on MSCOCO without any test-time overhead. Moreover, for the first time, especially on multi-person pose estimation, our regression method is superior to the heatmap-based methods. Our code is available at //github.com/Jeff-sjtu/res-loglikelihood-regression

There is growing interest in object detection in advanced driver assistance systems and autonomous robots and vehicles. To enable such innovative systems, we need faster object detection. In this work, we investigate the trade-off between accuracy and speed with domain-specific approximations, i.e. category-aware image size scaling and proposals scaling, for two state-of-the-art deep learning-based object detection meta-architectures. We study the effectiveness of applying approximation both statically and dynamically to understand the potential and the applicability of them. By conducting experiments on the ImageNet VID dataset, we show that domain-specific approximation has great potential to improve the speed of the system without deteriorating the accuracy of object detectors, i.e. up to 7.5x speedup for dynamic domain-specific approximation. To this end, we present our insights toward harvesting domain-specific approximation as well as devise a proof-of-concept runtime, AutoFocus, that exploits dynamic domain-specific approximation.

While Generative Adversarial Networks (GANs) have empirically produced impressive results on learning complex real-world distributions, recent work has shown that they suffer from lack of diversity or mode collapse. The theoretical work of Arora et al.~\cite{AroraGeLiMaZh17} suggests a dilemma about GANs' statistical properties: powerful discriminators cause overfitting, whereas weak discriminators cannot detect mode collapse. In contrast, we show in this paper that GANs can in principle learn distributions in Wasserstein distance (or KL-divergence in many cases) with polynomial sample complexity, if the discriminator class has strong distinguishing power against the particular generator class (instead of against all possible generators). For various generator classes such as mixture of Gaussians, exponential families, and invertible neural networks generators, we design corresponding discriminators (which are often neural nets of specific architectures) such that the Integral Probability Metric (IPM) induced by the discriminators can provably approximate the Wasserstein distance and/or KL-divergence. This implies that if the training is successful, then the learned distribution is close to the true distribution in Wasserstein distance or KL divergence, and thus cannot drop modes. Our preliminary experiments show that on synthetic datasets the test IPM is well correlated with KL divergence, indicating that the lack of diversity may be caused by the sub-optimality in optimization instead of statistical inefficiency.

In this paper, we propose an improved quantitative evaluation framework for Generative Adversarial Networks (GANs) on generating domain-specific images, where we improve conventional evaluation methods on two levels: the feature representation and the evaluation metric. Unlike most existing evaluation frameworks which transfer the representation of ImageNet inception model to map images onto the feature space, our framework uses a specialized encoder to acquire fine-grained domain-specific representation. Moreover, for datasets with multiple classes, we propose Class-Aware Frechet Distance (CAFD), which employs a Gaussian mixture model on the feature space to better fit the multi-manifold feature distribution. Experiments and analysis on both the feature level and the image level were conducted to demonstrate improvements of our proposed framework over the recently proposed state-of-the-art FID method. To our best knowledge, we are the first to provide counter examples where FID gives inconsistent results with human judgments. It is shown in the experiments that our framework is able to overcome the shortness of FID and improves robustness. Code will be made available.

We develop an approach to risk minimization and stochastic optimization that provides a convex surrogate for variance, allowing near-optimal and computationally efficient trading between approximation and estimation error. Our approach builds off of techniques for distributionally robust optimization and Owen's empirical likelihood, and we provide a number of finite-sample and asymptotic results characterizing the theoretical performance of the estimator. In particular, we show that our procedure comes with certificates of optimality, achieving (in some scenarios) faster rates of convergence than empirical risk minimization by virtue of automatically balancing bias and variance. We give corroborating empirical evidence showing that in practice, the estimator indeed trades between variance and absolute performance on a training sample, improving out-of-sample (test) performance over standard empirical risk minimization for a number of classification problems.

The Residual Networks of Residual Networks (RoR) exhibits excellent performance in the image classification task, but sharply increasing the number of feature map channels makes the characteristic information transmission incoherent, which losses a certain of information related to classification prediction, limiting the classification performance. In this paper, a Pyramidal RoR network model is proposed by analysing the performance characteristics of RoR and combining with the PyramidNet. Firstly, based on RoR, the Pyramidal RoR network model with channels gradually increasing is designed. Secondly, we analysed the effect of different residual block structures on performance, and chosen the residual block structure which best favoured the classification performance. Finally, we add an important principle to further optimize Pyramidal RoR networks, drop-path is used to avoid over-fitting and save training time. In this paper, image classification experiments were performed on CIFAR-10/100 and SVHN datasets, and we achieved the current lowest classification error rates were 2.96%, 16.40% and 1.59%, respectively. Experiments show that the Pyramidal RoR network optimization method can improve the network performance for different data sets and effectively suppress the gradient disappearance problem in DCNN training.