### Phasebook and Friends: Leveraging Discrete Representations for Source Separation

Deep learning based speech enhancement and source separation systems have recently reached unprecedented levels of quality, to the point that performance is reaching a new ceiling. Most systems rely on estimating the magnitude of a target source by estimating a real-valued mask to be applied to a time-frequency representation of the mixture signal. A limiting factor in such approaches is a lack of phase estimation: the phase of the mixture is most often used when reconstructing the estimated time-domain signal. Here, we propose MagBook', phasebook', and `Combook', three new types of layers based on discrete representations that can be used to estimate complex time-frequency masks. MagBook layers extend classical sigmoidal units and a recently introduced convex softmax activation for mask-based magnitude estimation. Phasebook layers use a similar structure to give an estimate of the phase mask without suffering from phase wrapping issues. Combook layers are an alternative to the MagBook-Phasebook combination that directly estimate complex masks. We present various training and inference regimes involving these representations, and explain in particular how to include them in an end-to-end learning framework. We also present an oracle study to assess upper bounds on performance for various types of masks using discrete phase representations. We evaluate the proposed methods on the wsj0-2mix dataset, a well-studied corpus for single-channel speaker-independent speaker separation, matching the performance of state-of-the-art mask-based approaches without requiring additional phase reconstruction steps.

### Phase Retrieval Under a Generative Prior

We introduce a novel deep-learning inspired formulation of the \textit{phase retrieval problem}, which asks to recover a signal $y_0 \in \R^n$ from $m$ quadratic observations, under structural assumptions on the underlying signal. As is common in many imaging problems, previous methodologies have considered natural signals as being sparse with respect to a known basis, resulting in the decision to enforce a generic sparsity prior. However, these methods for phase retrieval have encountered possibly fundamental limitations, as no computationally efficient algorithm for sparse phase retrieval has been proven to succeed with fewer than $O(k^2\log n)$ generic measurements, which is larger than the theoretical optimum of $O(k \log n)$. In this paper, we sidestep this issue by considering a prior that a natural signal is in the range of a generative neural network $G : \R^k \rightarrow \R^n$. We introduce an empirical risk formulation that has favorable global geometry for gradient methods, as soon as $m = O(k)$, under the model of a multilayer fully-connected neural network with random weights. Specifically, we show that there exists a descent direction outside of a small neighborhood around the true $k$-dimensional latent code and a negative multiple thereof. This formulation for structured phase retrieval thus benefits from two effects: generative priors can more tightly represent natural signals than sparsity priors, and this empirical risk formulation can exploit those generative priors at an information theoretically optimal sample complexity, unlike for a sparsity prior. We corroborate these results with experiments showing that exploiting generative models in phase retrieval tasks outperforms both sparse and general phase retrieval methods.

### Blind Deconvolutional Phase Retrieval via Convex Programming

We consider the task of recovering two real or complex $m$-vectors from phaseless Fourier measurements of their circular convolution. Our method is a novel convex relaxation that is based on a lifted matrix recovery formulation that allows a nontrivial convex relaxation of the bilinear measurements from convolution. We prove that if the two signals belong to known random subspaces of dimensions $k$ and $n$, then they can be recovered up to the inherent scaling ambiguity with $m >> (k+n) \log^2 m$ phaseless measurements. Our method provides the first theoretical recovery guarantee for this problem by a computationally efficient algorithm and does not require a solution estimate to be computed for initialization. Our proof is based Rademacher complexity estimates. Additionally, we provide an ADMM implementation of the method and provide numerical experiments that verify the theory.

### [Perspective] Photonic multitasking enabled with geometric phase

The constructive and destructive interference of waves is often exploited in optics and signal transmission. The interference pattern is a direct measure of the phase difference between two or more beams. Such a phase difference may result from the difference between the optical paths traversed by the light beams. However, phase can change for a single beam if it propagates through an "anisotropic parameter space," a medium that curves the light; this property is called geometric or topological phase (1–4). On page 1202 of this issue, Maguid et al. (5) use metasurfaces--ultrathin, planar engineered structures (6–9)--to form shared-aperture antenna arrays that impart geometric phase to optical signals.

### Orthogonal Matching Pursuit with Replacement

In this paper, we consider the problem of compressed sensing where the goal is to recover almost all the sparse vectors using a small number of fixed linear measurements. For this problem, we propose a novel partial hard-thresholding operator that leads to a general family of iterative algorithms. While one extreme of the family yields well known hard thresholding algorithms like ITI (Iterative Thresholding with Inversion) and HTP (Hard Thresholding Pursuit), the other end of the spectrum leads to a novel algorithm that we call Orthogonal Matching Pursuit with Replacement (OMPR). OMPR, like the classic greedy algorithm OMP, adds exactly one coordinate to the support at each iteration, based on the correlation with the current residual. However, unlike OMP, OMPR also removes one coordinate from the support. This simple change allows us to prove that OMPR has the best known guarantees for sparse recovery in terms of the Restricted Isometry Property (a condition on the measurement matrix). In contrast, OMP is known to have very weak performance guarantees under RIP. Given its simple structure, we are able to extend OMPR using locality sensitive hashing to get OMPR-Hash, the first provably sub-linear (in dimensionality) algorithm for sparse recovery. Our proof techniques are novel and flexible enough to also permit the tightest known analysis of popular iterative algorithms such as CoSaMP and Subspace Pursuit. We provide experimental results on large problems providing recovery for vectors of size up to million dimensions. We demonstrate that for large-scale problems our proposed methods are more robust and faster than existing methods.