Bispectrum Inversion with Application to Multireference Alignment.

Area of Science:

• Signal processing
• Computational mathematics
• Statistical inference

Background:

• Multireference alignment (MRA) is crucial for estimating signals from noisy, translated versions.
• Traditional MRA methods often involve estimating shifts before signal inference.
• Direct signal estimation offers an alternative approach to MRA.

Purpose of the Study:

• To develop and analyze algorithms for direct signal estimation in MRA.
• To leverage translation-invariant signal features for robust estimation.
• To investigate the performance of these methods under varying noise levels.

Main Methods:

• Estimating signal power spectrum and bispectrum from noisy observations.
• Utilizing non-convex optimization over the manifold of Fourier phases.
• Developing complementary methods: frequency marching, semidefinite relaxation, and integer programming.

Main Results:

• Non-convex optimization methods empirically converge to the true signal with random initialization in noise-free cases.
• Frequency marching and semidefinite relaxation provably recover phases exactly without noise.
• The invariant features approach provides stable estimation in high noise, achieving the information-theoretic rate.

Conclusions:

• Direct signal estimation using invariant features is a viable and robust strategy for MRA.
• The proposed algorithms demonstrate effectiveness across different noise regimes.
• The methods are efficient, requiring only one data pass, beneficial for low signal-to-noise ratio scenarios.