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. Humans are able to detect blurring of visual images, but the mechanism by which they do so is not clear. A traditional view is that a blurred image looks "unnatural" because of the reduction in energy (either globally or locally) at high frequencies. In this paper, we propose that the disruption of local phase can provide an alternative explanation for blur perception. We show that precisely localized features such as step edges result in strong local phase coherence structures across scale and space in the complex wavelet transform domain, and blurring causes loss of such phase coherence. We propose a technique for coarse-to-fine phase prediction of wavelet coefficients, and observe that (1) such predictions are highly effective in natural images, (2) phase coherence increases with the strength of image features, and (3) blurring disrupts the phase coherence relationship in images. We thus lay the groundwork for a new theory of perceptual blur estimation, as well as a variety of algorithms for restoration and manipulation of photographic images. A number of recent attempts have been made to describe early sensory coding in terms of a general information processing strategy. In this paper, two strategies are contrasted. Both strategies take advantage of the redundancy in the environment to produce more effective representations. The first is described as a "compact" coding scheme. A compact code performs a transform that allows the input to be represented with a reduced number of vectors (cells) with minimal RMS error. This approach has recently become popular in the neural network literature and is related to a process called Principal Components Analysis (PCA). A number of recent papers have suggested that the optimal compact code for representing natural scenes will have units with receptive field profiles much like those found in the retina and primary visual cortex. However, in this paper, it is proposed that compact coding schemes are insufficient to account for the receptive field properties of cells in the mammalian visual pathway. In contrast, it is proposed that the visual system is near to optimal in representing natural scenes only if optimality is defined in terms of "sparse distributed" coding. In a sparse distributed code, all cells in the code have an equal response probability across the class of images but have a low response probability for any single image. In such a code, the dimensionality is not reduced. Rather, the redundancy of the input is transformed into the redundancy of the firing pattern of cells. It is proposed that the signature for a sparse code is found in the fourth moment of the response distribution (i.e., the kurtosis). In measurements with 55 calibrated natural scenes, the kurtosis was found to peak when the bandwidths of the visual code matched those of cells in the mammalian visual cortex. Codes resembling "wavelet transforms" are proposed to be effective because the response histograms of such codes are sparse (i.e., show high kurtosis) when presented with natural scenes. It is proposed that the structure of the image that allows sparse coding is found in the phase spectrum of the image. It is suggested that natural scenes, to a first approximation, can be considered as a sum of self-similar local functions (the inverse of a wavelet). Possible reasons for why sensory systems would evolve toward sparse coding are presented. Humans, like other animals, are exposed to a continuous stream of signals, which are dynamic, multimodal, extended, and time varying in nature. This complex input space must be transduced and sampled by our sensory systems and transmitted to the brain where it can guide the selection of appropriate actions. To simplify this process, it's been suggested that the brain exploits statistical regularities in the stimulus space. Tests of this idea have largely been confined to unimodal signals and natural scenes. One important class of multisensory signals for which a quantitative input space characterization is unavailable is human speech. We do not understand what signals our brain has to actively piece together from an audiovisual speech stream to arrive at a percept versus what is already embedded in the signal structure of the stream itself. In essence, we do not have a clear understanding of the natural statistics of audiovisual speech. In the present study, we identified the following major statistical features of audiovisual speech. First, we observed robust correlations and close temporal correspondence between the area of the mouth opening and the acoustic envelope. Second, we found the strongest correlation between the area of the mouth opening and vocal tract resonances. Third, we observed that both area of the mouth opening and the voice envelope are temporally modulated in the 2–7 Hz frequency range. Finally, we show that the timing of mouth movements relative to the onset of the voice is consistently between 100 and 300 ms. We interpret these data in the context of recent neural theories of speech which suggest that speech communication is a reciprocally coupled, multisensory event, whereby the outputs of the signaler are matched to the neural processes of the receiver.