Bispectrum Inversion with Application to Multireference Alignment

By Tamir Bendory, Nicolas Boumal, Chao Ma, Zhizhen Zhao, Amit Singer. 2017

107 citations recorded by the source corpus.

We consider the problem of estimating a signal from noisy circularly translated versions of itself, called multireference alignment (MRA). One natural approach to MRA could be to estimate the shifts of the observations first, and infer the signal by aligning and averaging the data. In contrast, we consider a method based on estimating the signal directly, using features of the signal that are invariant under translations. Specifically, we estimate the power spectrum and the bispectrum of the signal from the observations. Under mild assumptions, these invariant features contain enough information to infer the signal. In particular, the bispectrum can be used to estimate the Fourier phases. To this end, we propose and analyze a few algorithms. Our main methods consist of nonconvex optimization over the smooth manifold of phases. Empirically, in the absence of noise, these nonconvex algorithms appear to converge to the target signal with random initialization. The algorithms are also robust to noise. We then suggest three additional methods. These methods are based on frequency marching, semidefinite relaxation, and integer programming. The first two methods provably recover the phases exactly in the absence of noise. In the high noise level regime, the invariant features approach for MRA results in stable estimation if the number of measurements scales like the cube of the noise variance, which is the information-theoretic rate. Additionally, it requires only one pass over the data, which is important at low signal-to-noise ratio when the number of observations must be large.

Subfields: Computational Mechanics, Computer Vision and Pattern Recognition, Signal Processing

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Citations in the corpus

Cormorant: Covariant Molecular Neural Networks (2019, methodology): “Note the similarity of this expression to the bispectrum [Kakarala, 1992, Bendory et al., 2018], which is an already established tool in the force ﬁeld learning literature [Bartók et al., 2013].”
A Group-Theoretic Framework for Data Augmentation (2019, background): “For example, a line of work proposes to add random or adversarial noise when training [11, 51, 26, 74, 8, 49, 73, 37] and argue that this can lead to a form of regularization [63, 34, 35, 18, 71].”