In this task, you are given a part of an article. Your task is to generate headline (title) for this text. Preferred headlines are under fifteen words.

Ex Input:
Rough set theory, a mathematical tool to deal with vague concepts, has originally described the indiscernibility of elements by equivalence relations. Covering rough sets are a natural extension of classical rough sets by relaxing the partitions arising from equivalence relations to covers. Recently, some topological concepts such as neighborhood have been applied to covering rough sets. In this paper, we further investigate the covering rough sets based on neighborhoods by approximation operations. We show that the upper approximation based on neighborhoods can be defined equivalently without using neighborhoods. To analyze the covers themselves, we introduce unary and composition operations on covers. A notion of homomorphism is provided to relate two covering approximation spaces. We also examine the properties of approximations preserved by the operations and homomorphisms, respectively.

Ex Output:
Covering rough sets based on neighborhoods

Ex Input:
Sliding window convolutional networks (ConvNets) have become a popular approach to computer vision problems such as image segmentation, and object detection and localization. Here we consider the problem of inference, the application of a previously trained ConvNet, with emphasis on 3D images. Our goal is to maximize throughput, defined as average number of output voxels computed per unit time. Other things being equal, processing a larger image tends to increase throughput, because fractionally less computation is wasted on the borders of the image. It follows that an apparently slower algorithm may end up having higher throughput if it can process a larger image within the constraint of the available RAM. We introduce novel CPU and GPU primitives for convolutional and pooling layers, which are designed to minimize memory overhead. The primitives include convolution based on highly efficient pruned FFTs. Our theoretical analyses and empirical tests reveal a number of interesting findings. For some ConvNet architectures, cuDNN is outperformed by our FFT-based GPU primitives, and these in turn can be outperformed by our CPU primitives. The CPU manages to achieve higher throughput because of its fast access to more RAM. A novel primitive in which the GPU accesses host RAM can significantly increase GPU throughput. Finally, a CPU-GPU algorithm achieves the greatest throughput of all, 10 or more than other publicly available implementations of sliding window 3D ConvNets. All of our code has been made available as open source project.

Ex Output:
ZNNi Maximizing the Inference Throughput of 3D Convolutional Networks on Multi-Core CPUs and GPUs

Ex Input:
Deep learning tools have recently gained much attention in applied machine learning. However such tools for regression and classification do not allow us to capture model uncertainty. Bayesian models offer us the ability to reason about model uncertainty, but usually come with a prohibitive computational cost. We show that dropout in multilayer perceptron models (MLPs) can be interpreted as a Bayesian approximation. Results are obtained for modelling uncertainty for dropout MLP models extracting information that has been thrown away so far, from existing models. This mitigates the problem of representing uncertainty in deep learning without sacrificing computational performance or test accuracy. We perform an exploratory study of the dropout uncertainty properties. Various network architectures and non-linearities are assessed on tasks of extrapolation, interpolation, and classification. We show that model uncertainty is important for classification tasks using MNIST as an example, and use the model's uncertainty in a Bayesian pipeline, with deep reinforcement learning as a concrete example.

Ex Output:
Dropout as a Bayesian Approximation: Representing Model Uncertainty in Deep Learning