Wavelet invariants for statistically robust multi-reference alignment. (13th August 2020)

• Record Type: Journal Article
• Authors: Hirn, Matthew; Little, Anna
• Is Part Of: Information and inference. Volume 10:Number 4(2021)
• Page Start: 1287
• Page End: 1351
• Publication Date: 2020-08-13
• Subjects: multi-reference alignment -- method of invariants -- wavelets -- signal processing -- wavelet scattering transform; Mathematical models -- Periodicals; 519.605
• DOI: 10.1093/imaiai/iaaa016
• Languages: English
• ISSNs: 2049-8764

Abstract

Abstract: We propose a nonlinear, wavelet-based signal representation that is translation invariant and robust to both additive noise and random dilations. Motivated by the multi-reference alignment problem and generalizations thereof, we analyze the statistical properties of this representation given a large number of independent corruptions of a target signal. We prove the nonlinear wavelet-based representation uniquely defines the power spectrum but allows for an unbiasing procedure that cannot be directly applied to the power spectrum. After unbiasing the representation to remove the effects of the additive noise and random dilations, we recover an approximation of the power spectrum by solving a convex optimization problem, and thus reduce to a phase retrieval problem. Extensive numerical experiments demonstrate the statistical robustness of this approximation procedure.