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.

Q: We propose a solution for evaluation of mathematical expression. However, instead of designing a single end-to-end model we propose a Lego bricks style architecture. In this architecture instead of training a complex end-to-end neural network, many small networks can be trained independently each accomplishing one specific operation and acting a single lego brick. More difficult or complex task can then be solved using a combination of these smaller network. In this work we first identify 8 fundamental operations that are commonly used to solve arithmetic operations (such as 1 digit multiplication, addition, subtraction, sign calculator etc). These fundamental operations are then learned using simple feed forward neural networks. We then shows that different operations can be designed simply by reusing these smaller networks. As an example we reuse these smaller networks to develop larger and a more complex network to solve n-digit multiplication, n-digit division, and cross product. This bottom-up strategy not only introduces reusability, we also show that it allows to generalize for computations involving n-digits and we show results for up to 7 digit numbers. Unlike existing methods, our solution also generalizes for both positive as well as negative numbers.

A: We train many small networks each for a specific operation, these are then combined to perform complex operations
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Q: Audio signals are sampled at high temporal resolutions, and learning to synthesize audio requires capturing structure across a range of timescales. Generative adversarial networks (GANs) have seen wide success at generating images that are both locally and globally coherent, but they have seen little application to audio generation. In this paper we introduce WaveGAN, a first attempt at applying GANs to unsupervised synthesis of raw-waveform audio. WaveGAN is capable of synthesizing one second slices of audio waveforms with global coherence, suitable for sound effect generation. Our experiments demonstrate that—without labels—WaveGAN learns to produce intelligible words when trained on a small-vocabulary speech dataset, and can also synthesize audio from other domains such as drums, bird vocalizations, and piano. We compare WaveGAN to a method which applies GANs designed for image generation on image-like audio feature representations, finding both approaches to be promising.

A: Learning to synthesize raw waveform audio with GANs
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Q: 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.

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