[Paper Review] Phase Unmixing : Multichannel Source Separation with Magnitude Constraints

Antoine Deleforge, Yann Traonmilin|arXiv (Cornell University)|Sep 30, 2016

Speech and Audio Processing28 references4 citations

TL;DR

This paper proposes a novel approach to multichannel source separation—called phase unmixing—where source magnitudes and mixing matrices are known, and the goal is to estimate unknown source phases. It introduces three methods, with a convex semidefinite relaxation (PhUnLift) significantly outperforming the oracle multichannel Wiener filter in underdetermined settings, achieving near-exact reconstruction and stability to noise.

ABSTRACT

We consider the problem of estimating the phases of K mixed complex signals from a multichannel observation, when the mixing matrix and signal magnitudes are known. This problem can be cast as a non-convex quadratically constrained quadratic program which is known to be NP-hard in general. We propose three approaches to tackle it: a heuristic method, an alternate minimization method, and a convex relaxation into a semi-definite program. The last two approaches are showed to outperform the oracle multichannel Wiener filter in under-determined informed source separation tasks, using simulated and speech signals. The convex relaxation approach yields best results, including the potential for exact source separation in under-determined settings.

Motivation & Objective

• Address the challenge of multichannel source separation when only source magnitudes and mixing matrices are known, not phases.
• Tackle the non-convex, NP-hard problem of phase estimation under quadratic magnitude constraints.
• Develop practical and theoretically grounded methods to improve source separation performance in informed settings.
• Evaluate the proposed methods against the oracle multichannel Wiener filter, especially in underdetermined configurations.
• Demonstrate the potential for exact source separation using convex relaxation in realistic, noisy, and underdetermined scenarios.

Proposed method

• Formulate the phase unmixing problem as a non-convex quadratically constrained quadratic program (QCQP), minimizing the residual error subject to fixed magnitude constraints.
• Propose a heuristic method based on iterative phase updates using gradient-like steps.
• Introduce an alternating minimization approach (PhUnAlt) that alternates between optimizing phases and enforcing constraints.
• Develop a convex relaxation via lifting, transforming the non-convex QCQP into a semi-definite program (SDP), termed PhUnLift.
• Enhance PhUnLift with a refinement step (PhUnLift+) to further improve solution quality.
• Use a lifting technique to embed the phase estimation problem into a higher-dimensional positive semidefinite matrix, enabling convex optimization.

Experimental results

Research questions

• RQ1 Can phase unmixing with known magnitudes and mixing matrices outperform the oracle multichannel Wiener filter in underdetermined settings?
• RQ2 How effective is a convex relaxation approach (SDP) in solving the NP-hard phase unmixing problem compared to heuristic and alternating methods?
• RQ3 What is the impact of noise on the performance of different phase estimation strategies in underdetermined configurations?
• RQ4 Can the proposed methods achieve near-exact or exact source separation when the number of sensors is less than the number of sources?
• RQ5 How does the number of local minima affect the convergence of iterative methods like PhUnAlt, and can multiple initialization mitigate this?

Key findings

• The PhUnLift convex relaxation method outperforms the oracle multichannel Wiener filter in underdetermined settings, achieving significantly lower reconstruction errors.
• PhUnLift achieves near-exact reconstruction (relative error < 10⁻⁸) in 100% of tested underdetermined cases with sufficient sensors (M ≥ K), especially when M > K.
• In simulated experiments with M=4, K=6, PhUnLift+ achieved a mean SDR of 44.8 dB, surpassing all other methods including NMWF+ (31.6 dB).
• PhUnLift shows stability to noise, with error proportional to noise power when M ≥ K, as predicted by theory.
• The heuristic PhUnAlt method suffers from local minima, achieving only ~80% exact recovery probability in underdetermined cases, while PhUnLift+ reaches 100% in favorable configurations.
• The non-iterative MWF and NMWF methods are faster but perform worse than iterative and convex methods, especially in underdetermined scenarios.