Fourier methods have a long and proven track record as an excellent tool in data processing. We propose to integrate Fourier methods into complex recurrent neural network architectures and show accuracy improvements on prediction tasks as well as computational load reductions. We predict synthetic data drawn from synthetic-equations as well as real world power load data.
Recurrent neural networks (RNNs) are notoriously difficult to train. When the eigenvalues of the hidden to hidden weight matrix deviate from absolute value 1, optimization becomes difficult due to the well studied issue of vanishing and exploding gradients, especially when trying to learn long-term dependencies. To circumvent this problem, we propose a new architecture that learns a unitary weight matrix, with eigenvalues of absolute value exactly 1. The challenge we address is that of parametrizing unitary matrices in a way that does not require expensive computations (such as eigendecomposition) after each weight update. We construct an expressive unitary weight matrix by composing several structured matrices that act as building blocks with parameters to be learned. Optimization with this parameterization becomes feasible only when considering hidden states in the complex domain. We demonstrate the potential of this architecture by achieving state of the art results in several hard tasks involving very longterm dependencies. We present Listen, Attend and Spell (LAS), a neural speech recognizer that transcribes speech utterances directly to characters without pronunciation models, HMMs or other components of traditional speech recognizers. In LAS, the neural network architecture subsumes the acoustic, pronunciation and language models making it not only an end-to-end trained system but an end-to-end model. In contrast to DNN-HMM, CTC and most other models, LAS makes no independence assumptions about the probability distribution of the output character sequences given the acoustic sequence. Our system has two components: a listener and a speller. The listener is a pyramidal recurrent network encoder that accepts filter bank spectra as inputs. The speller is an attention-based recurrent network decoder that emits each character conditioned on all previous characters, and the entire acoustic sequence. On a Google voice search task, LAS achieves a WER of 14.1 without a dictionary or an external language model and 10.3 with language model rescoring over the top 32 beams. In comparison, the state-of-the-art CLDNN-HMM model achieves a WER of 8.0 on the same set. At present, the vast majority of building blocks, techniques, and architectures for deep learning are based on real-valued operations and representations. However, recent work on recurrent neural networks and older fundamental theoretical analysis suggests that complex numbers could have a richer representational capacity and could also facilitate noise-robust memory retrieval mechanisms. Despite their attractive properties and potential for opening up entirely new neural architectures, complex-valued deep neural networks have been marginalized due to the absence of the building blocks required to design such models. In this work, we provide the key atomic components for complex-valued deep neural networks and apply them to convolutional feed-forward networks. More precisely, we rely on complex convolutions and present algorithms for complex batch-normalization, complex weight initialization strategies for complex-valued neural nets and we use them in experiments with end-to-end training schemes. We demonstrate that such complex-valued models are competitive with their real-valued counterparts. We test deep complex models on several computer vision tasks, on music transcription using the MusicNet dataset and on Speech spectrum prediction using TIMIT. We achieve state-of-the-art performance on these audio-related tasks.   This paper introduces a new large-scale music dataset, MusicNet, to serve as a source of supervision and evaluation of machine learning methods for music research. MusicNet consists of hundreds of freely-licensed classical music recordings by 10 composers, written for 11 instruments, together with instrument note annotations resulting in over 1 million temporal labels on 34 hours of chamber music performances under various studio and microphone conditions. @PARASPLIT The paper defines a multi-label classification task to predict notes in musical recordings, along with an evaluation protocol, and benchmarks several machine learning architectures for this task: i) learning from spectrogram features; ii) end-to-end learning with a neural net; iii) end-to-end learning with a convolutional neural net. These experiments show that end-to-end models trained for note prediction learn frequency selective filters as a low-level representation of audio.