Phase Retrieval With Bregman Divergences and Application to Audio Signal Recovery

Abstract

Phase retrieval (PR) aims to recover a signal from the magnitudes of a set of inner products. This problem arises in many audio signal processing applications which operate on a short-time Fourier transform magnitude or power spectrogram, and discard the phase information. Recovering the missing phase from the resulting modified spectrogram is indeed necessary in order to synthesize time-domain signals. PR is commonly addressed by considering a minimization problem involving a quadratic loss function. In this paper, we adopt a different standpoint. Indeed, the quadratic loss does not properly account for some perceptual properties of audio, and alternative discrepancy measures such as beta-divergences have been preferred in many settings. Therefore, we formulate PR as a new minimization problem involving Bregman divergences. Since these divergences are not symmetric with respect to their two input arguments in general, they lead to two different formulations of the problem. To optimize the resulting objective, we derive two algorithms based on accelerated gradient descent and alternating direction method of multipliers. Experiments conducted on audio signal recovery from spectrograms that are either exact or estimated from noisy observations highlight the potential of our proposed methods for audio restoration. In particular, leveraging some of these Bregman divergences induce better performance than the quadratic loss when performing PR from spectrograms under very noisy conditions.

Phase recovery with Bregman divergences for audio source separation

“…where A † is the Moore-Penrose pseudo-inverse of A defined as A † = (A H A) .−1 A H , which encodes the inverse STFT. This iterative scheme, known as the Griffin-Lim (GL) algorithm, is proved to converge to a critical point of the quadratic loss in (1) [19], and can also be obtained by majorization-minimization [20] or using a gradient descent scheme [17]. Improvements of this algorithm notably include accelerated [21] and real-time purposed versions [22].…”

“…In [17], we proposed to replace the quadratic loss in problem (1) with Bregman divergences, which encompass the β-divergence [15] and its special cases, the KL and IS divergences. A Bregman divergence D ψ is defined from a strictly-convex, continuously-differentiable generating function ψ (with derivative ψ ′ ) as follows:…”

“…Typical Bregman divergences with their generating function and derivative can be found, e.g., in [17] (see Table 1).…”

“…In [17] we derived two algorithms for solving both problems, based on gradient descent and alternating direction method of multipliers (ADMM).…”

“…These divergences are acknowledged for their superior performance in audio spectral decomposition applications such as NMF-based source separation [16]. In a previous work [17], we addressed phase recovery with the Bregman divergences in a single-source setting. Here, we propose to extend this approach to a single-channel and multiple-sources framework, where the mixture's information can be exploited.…”