A Spectral Method for Stable Bispectrum Inversion With Application to Multireference Alignment

Chen, Hua; Zehni, Mona; Zhao, Zhizhen

doi:10.1109/lsp.2018.2831631

Cited by 21 publications

(16 citation statements)

References 24 publications

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“…In fact, no method can succeed with asymptotically fewer measurements (as a function of the SNR) in the case of uniform distribution of shifts [70][71][72]. The computational complexity and stability of a variety of bispectrum inversion algorithms was studied in [73,74]. A somewhat surprising result is that multi-reference alignment with non-uniform (more precisely, non-periodic) distribution of shifts can be solved with just the first two moments and the sample complexity is proportional to 1/SNR 2 [75,76].…”

Section: G a Mathematical Toy Model: Multi-reference Alignmentmentioning

confidence: 99%

Mathematics for Cryo-Electron Microscopy

Singer

2019

Proceedings of the International Congress of Mathematicians (ICM 2018)

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Single-particle cryo-electron microscopy (cryo-EM) has recently joined X-ray crystallography and NMR spectroscopy as a high-resolution structural method for biological macromolecules. Cryo-EM was selected by Nature Methods as Method of the Year 2015, large scale investments in cryo-EM facilities are being made all over the world, and the Nobel Prize in Chemistry 2017 was awarded to Jacques Dubochet, Joachim Frank and Richard Henderson "for developing cryoelectron microscopy for the high-resolution structure determination of biomolecules in solution". This paper focuses on the mathematical principles underlying existing algorithms for structure determination using single particle cryo-EM. A IntroductionThe field of structural biology is currently undergoing a transformative change [1,2]. Structures of many biomolecular targets previously insurmountable by X-ray crystallography are now being obtained using single particle cryo-EM to resolutions beyond 4Å on a regular basis [3][4][5]. This leap in cryo-EM technology, as recognized by the 2017 Nobel Prize in Chemistry, is mainly due to hardware advancements including the invention of the direct electron detector and the methodological development of algorithms for data processing. Cryo-EM is a very general and powerful technique because it does not require the formation of crystalline arrays of macromolecules. In addition, unlike X-ray crystallography and nuclear magnetic resonance (NMR) that measure ensembles of particles, single particle cryo-EM produces images of individual particles. Cryo-EM therefore has the potential to analyze conformational changes and energy landscapes associated with structures of complexes in different functional states.The main purpose of this brief review paper is to expose mathematicians to the exciting field of cryo-EM. As there exist many excellent review articles and textbooks on single particle cryo-EM [6-12], we choose to solely focus here on the mathematical foundations of this technique. Topics of great importance to practitioners, such as the physics and optics of the electron microscope, sample preparation, and data acquisition are not treated here.In cryo-EM, biological macromolecules are imaged in an electron microscope. The molecules are rapidly frozen in a thin layer of vitreous ice, trapping them in a nearly-physiological state. The molecules are randomly oriented and positioned within the ice layer. The electron microscope produces a two-dimensional tomographic projection image (called a micrograph) of the molecules embedded in the ice layer. More specifically, what is being measured by the detector is the integral in the direction of the beaming electrons of the electrostatic potential of the individual molecules.

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Section: G a Mathematical Toy Model: Multi-reference Alignmentmentioning

confidence: 99%

Mathematics for Cryo-Electron Microscopy

Singer

2019

Proceedings of the International Congress of Mathematicians (ICM 2018)

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“…These features constitute the moments of the signal and are estimated from the measurements. The signal is then estimated from the features via an optimization-based framework [8,11], tensor decomposition [12,13] using Jennrich's algorithm [14] or spectral decomposition [15,16]. As these works are specialized for MRA, they do not address the challenges associated with MSR, such as observing only shorter segments of the signal.…”

Section: Introductionmentioning

confidence: 99%

MSR-GAN: Multi-Segment Reconstruction via Adversarial Learning

Zehni

Zhao

2021

ICASSP 2021 - 2021 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)

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Multi-segment reconstruction (MSR) is the problem of estimating a signal given noisy partial observations. Here each observation corresponds to a randomly located segment of the signal. While previous works address this problem using template or momentmatching, in this paper we address MSR from an unsupervised adversarial learning standpoint, named MSR-GAN. We formulate MSR as a distribution matching problem where the goal is to recover the signal and the probability distribution of the segments such that the distribution of the generated measurements following a known forward model is close to the real observations. This is achieved once a min-max optimization involving a generator-discriminator pair is solved. MSR-GAN is mainly inspired by CryoGAN [1]. However, in MSR-GAN we no longer assume the probability distribution of the latent variables, i.e. segment locations, is given and seek to recover it alongside the unknown signal. For this purpose, we show that the loss at the generator side originally is non-differentiable with respect to the segment distribution. Thus, we propose to approximate it using Gumbel-Softmax reparametrization trick. Our proposed solution is generalizable to a wide range of inverse problems. Our simulation results and comparison with various baselines verify the potential of our approach in different settings.

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“…One is first estimating shifts and then estimating the true signal [2]. the other is estimating the true signal without seeking for shifts [15,32]. Here we are only concentrated on the latter.…”

Section: Introductionmentioning

confidence: 99%

An adaptive variational model for multireference alignment with mixed noise

Zhao¹,

Li²,

Gong³

2021

Preprint

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Multireference alignment (MRA) problem is to estimate an underlying signal from a large number of noisy circularly-shifted observations. The existing methods are always proposed under the hypothesis of a single Gaussian noise. However, the hypothesis of a single-type noise is inefficient for solving practical problems like single particle cryo-EM. In this paper, We focus on the MRA problem under the assumption of Gaussian mixture noise. We derive an adaptive variational model by combining maximum a posteriori (MAP) estimation and soft-max method. There are two adaptive weights which are for detecting cyclical shifts and types of noise. Furthermore, we provide a statistical interpretation of our model by using expectation-maximization(EM) algorithm. The existence of a minimizer is mathematically proved. The numerical results show that the proposed model has a more impressive performance than the existing methods when one Gaussian noise is large and the other is small.

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A Spectral Method for Stable Bispectrum Inversion With Application to Multireference Alignment

Cited by 21 publications

References 24 publications

Mathematics for Cryo-Electron Microscopy

Mathematics for Cryo-Electron Microscopy

MSR-GAN: Multi-Segment Reconstruction via Adversarial Learning

An adaptive variational model for multireference alignment with mixed noise

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