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 and convolutional LSTMs. 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 the TIMIT dataset. We achieve state-of-the-art performance on these audio-related tasks.
The phase component is not only important from a biological point of view but also from a signal processing perspective. It has been shown that the phase information in speech signals affects their intelligibility . Also @cite_6 show that the amount of information present in the phase of an image is sufficient to recover the majority of the information encoded in its magnitude. In fact, phase provides a detailed description of objects as it encodes shapes, edges, and orientations.
@cite_6
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In the Fourier representation of signals, spectral magnitude and phase tend to play different roles and in some situations many of the important features of a signal are preserved if only the phase is retained. Furthermore, under a variety of conditions, such as when a signal is of finite length, phase information alone is sufficient to completely reconstruct a signal to within a scale factor. In this paper, we review and discuss these observations and results in a number of different contexts and applications. Specifically, the intelligibility of phase-only reconstruction for images, speech, and crystallographic structures are illustrated. Several approaches to justifying the relative importance of phase through statistical arguments are presented, along with a number of informal arguments suggesting reasons for the importance of phase. Specific conditions under which a sequence can be exactly reconstructed from phase are reviewed, both for one-dimensional and multi-dimensional sequences, and algorithms for both approximate and exact reconstruction of signals from phase information are presented. A number of applications of the observations and results in this paper are suggested.