In this task, you are given the abstract of a research paper. Your task is to generate a summary of this abstract. Your summary should not be very short, but it's better if it's not more than 30 words.

Ex Input:
Neural networks are known to be a class of highly expressive functions able to fit even random input-output mappings with 100% accuracy. In this work we present properties of neural networks that complement this aspect of expressivity. By using tools from Fourier analysis, we show that deep ReLU networks are biased towards low frequency functions, meaning that they cannot have local fluctuations without affecting their global behavior. Intuitively, this property is in line with the observation that over-parameterized networks find simple patterns that generalize across data samples. We also investigate how the shape of the data manifold affects expressivity by showing evidence that learning high frequencies gets easier with increasing manifold complexity, and present a theoretical understanding of this behavior. Finally, we study the robustness of the frequency components with respect to parameter perturbation, to develop the intuition that the parameters must be finely tuned to express high frequency functions.

Ex Output:
We investigate ReLU networks in the Fourier domain and demonstrate peculiar behaviour.

Ex Input:
Weight pruning has proven to be an effective method in reducing the model size and computation cost while not sacrificing the model accuracy. Conventional sparse matrix formats, however, involve irregular index structures with large storage requirement and sequential reconstruction process, resulting in inefficient use of highly parallel computing resources. Hence, pruning is usually restricted to inference with a batch size of one, for which an efficient parallel matrix-vector multiplication method exists. In this paper, a new class of sparse matrix representation utilizing Viterbi algorithm that has a high, and more importantly, fixed index compression ratio regardless of the pruning rate, is proposed. In this approach, numerous sparse matrix candidates are first generated by the Viterbi encoder, and then the one that aims to minimize the model accuracy degradation is selected by the Viterbi algorithm. The model pruning process based on the proposed Viterbi encoder and Viterbi algorithm is highly parallelizable, and can be implemented efficiently in hardware to achieve low-energy, high-performance index decoding process. Compared with the existing magnitude-based pruning methods, index data storage requirement can be further compressed by 85.2% in MNIST and 83.9% in AlexNet while achieving similar pruning rate. Even compared with the relative index compression technique, our method can still reduce the index storage requirement by 52.7% in MNIST and 35.5% in AlexNet.

Ex Output:
We present a new pruning method and sparse matrix format to enable high index compression ratio and parallel index decoding process.

Ex Input:
Graph neural networks have recently achieved great successes in predicting quantum mechanical properties of molecules. These models represent a molecule as a graph using only the distance between atoms (nodes) and not the spatial direction from one atom to another. However, directional information plays a central role in empirical potentials for molecules, e.g. in angular potentials. To alleviate this limitation we propose directional message passing, in which we embed the messages passed between atoms instead of the atoms themselves. Each message is associated with a direction in coordinate space. These directional message embeddings are rotationally equivariant since the associated directions rotate with the molecule. We propose a message passing scheme analogous to belief propagation, which uses the directional information by transforming messages based on the angle between them. Additionally, we use spherical Bessel functions to construct a theoretically well-founded, orthogonal radial basis that achieves better performance than the currently prevalent Gaussian radial basis functions while using more than 4x fewer parameters. We leverage these innovations to construct the directional message passing neural network (DimeNet). DimeNet outperforms previous GNNs on average by 77% on MD17 and by 41% on QM9.

Ex Output:
Directional message passing incorporates spatial directional information to improve graph neural networks.