Recent advances have made it possible to create deep complex-valued neural networks. Despite this progress, the potential power of fully complex intermediate computations and representations has not yet been explored for many challenging learning problems. Building on recent advances, we propose a novel mechanism for extracting signals in the frequency domain. As a case study, we perform audio source separation in the Fourier domain. Our extraction mechanism could be regarded as a local ensembling method that combines a complex-valued convolutional version of Feature-Wise Linear Modulation (FiLM) and a signal averaging operation. We also introduce a new explicit amplitude and phase-aware loss, which is scale and time invariant, taking into account the complex-valued components of the spectrogram. Using the Wall Street Journal Dataset, we compare our phase-aware loss to several others that operate both in the time and frequency domains and demonstrate the effectiveness of our proposed signal extraction method and proposed loss. When operating in the complex-valued frequency domain, our deep complex-valued network substantially outperforms its real-valued counterparts even with half the depth and a third of the parameters. Our proposed mechanism improves significantly deep complex-valued networks' performance and we demonstrate the usefulness of its regularizing effect. Complex-valued neural networks have been studied since long before the emergence of modern deep learning techniques (Georgiou & Koutsougeras, 1992; Zemel et al., 1995; Kim & Adalı, 2003; Hirose, 2003; Nitta, 2004) . Nevertheless, deep complex-valued models have only started to gain momentum (Reichert & Serre, 2014; Arjovsky et al., 2015; Danihelka et al., 2016; Trabelsi et al., 2017; Jose et al., 2017; Wolter & Yao, 2018b; Choi et al., 2019) , with the great majority of models in deep learning still relying on real-valued representations. The motivation for using complex-valued representations for deep learning is twofold: On the one hand, biological nervous systems actively make use of synchronization effects to gate signals between neurons -a mechanism that can be recreated in artificial systems by taking into account phase differences (Reichert & Serre, 2014) . On the other hand, complex-valued representations are better suited to certain types of data, particularly those that are naturally expressed in the frequency domain. Other benefits provided by working with complex-valued inputs in the spectral or frequency domain are computational. In particular, short-time Fourier transforms (STFTs) can be used to considerably reduce the temporal dimension of the representation for an underlying signal. This is a critical advantage, as training recurrent neural networks (RNNs) or convolutional neural networks (CNNs) on long sequences remains challenging due to unstable gradients and the computational requirements of backpropagation through time (BPTT) (Hochreiter, 1991; Bengio et al., 1994) . Applying the STFT on the raw signal, on the other hand, is computationally efficient, as in practice it is implemented with the fast Fourier transform (FFT) whose computational complexity is O(n log(n)). The aforementioned biological, representational and computational considerations provide compelling motivations for designing learning models for tasks where the complex-valued representation of the input and output data is more desirable than their real-counterpart. Recent work has provided building blocks for deep complex-valued neural networks (Trabelsi et al., 2017) . These building blocks have been shown, in many cases, to avoid numerical problems during training and, thereby, enable the use of complex-valued representations. These representations are well-suited for frequency domain signals, as they have the ability to explicitly encode frequency magnitude and phase components. This motivates us to design a new signal extraction mechanism operating in the frequency domain. In this work, our contributions are summarized as follows: 1. We present a new signal separation mechanism implementing a local ensembling procedure. More precisely, a complex-valued convolutional version of Feature-wise Linear Modulation (FiLM) (Perez et al., 2018 ) is used to create multiple separated candidates for each of the signals we aim to retrieve from a mixture of inputs. A signal averaging operation on the candidates is then performed in order to increase the robustness of the signal to noise and interference. Before the averaging procedure, a form of dropout is implemented on the signal candidates in order to reduce the amount of interference and noise correlation existing between the different candidates. 2. We propose and explore a new magnitude and phase-aware loss taking explicitly into account the magnitude and phase of signals. A key characteristic of our loss is that it is scale-and time-invariant. We test our proposed signal extraction mechanism in the audio source separation setting where we aim to retrieve distinct audio signals associated with each speaker in the input mix. Our experiments demonstrate the usefulness of our extraction method, and show its regularizing effect. In this work, we introduced a new complex-valued extraction mechanism for signal retrieval in the Fourier domain. As a case study, we considered audio source separation. We also proposed a new phase-aware loss taking, explicitly, into account the magnitude and phase of the reference and estimated signals. The amplitude and phase-aware loss improves over other frequency and time-domain losses. We believe that our proposed method could lead to new research directions where signal retrieval is needed. A APPENDIX
New Signal Extraction Method in the Fourier Domain Contributes a complex-valued convolutional version of the Feature-Wise Linear Modulation which allows parameter optimization and designs a loss which takes into account magnitude and phase.