Multireference Alignment is Easier with an Aperiodic Translation Distribution

Abbé, Emmanuel; Bendory, Tamir; Leeb, William; Pereira, J. S.; Sharon, Nir; Singer, Amit

doi:10.48550/arxiv.1710.02793

Cited by 6 publications

(35 citation statements)

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“…First, (10) is not numerically stable: if z[0] or z[L − 1] are close to 0, recovery of z is sensitive to errors in the autocorrelations. In practice, we recover z by fitting it to its autocorrelations using a nonconvex least-squares (LS) procedure, which is empirically more robust to additive noise; we have observed similar phenomena for related problems [5,9,1]. Second, note that the second-order autocorrelation is not by itself sufficient to determine the signal uniquely.…”

Section: Recovering a Signal From Autocorrelationsmentioning

confidence: 90%

Toward single particle reconstruction without particle picking: Breaking the detection limit

Bendory

Boumal

Leeb

et al. 2018

Preprint

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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. In a cryo-EM experiment, the microscope produces images called micrographs. Projections of the molecule of interest are embedded in the micrographs at unknown locations, and under unknown viewing directions. Standard imaging techniques rst locate these projections (detection) and then reconstruct the 3-D structure from them. Unfortunately, high noise levels hinder detection. When reliable detection is rendered impossible, the standard techniques fail. This is a problem especially for small molecules, which can be particularly hard to detect. In this paper, we propose a radically dierent approach: we contend that the structure could, in principle, be reconstructed directly from the micrographs, without intermediate detection. As a result, even small molecules should be within reach for cryo-EM. To support this claim, we setup a simplied mathematical model and demonstrate how our autocorrelation analysis technique allows to go directly from the micrographs to the sought signals. This involves only one pass over the micrographs, which is desirable for large experiments. We show numerical results and discuss challenges that lay ahead to turn this proof-of-concept into a competitive alternative to state-of-the-art algorithms.

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Section: Recovering a Signal From Autocorrelationsmentioning

confidence: 90%

Toward single particle reconstruction without particle picking: Breaking the detection limit

Bendory

Boumal

Leeb

et al. 2018

Preprint

Self Cite

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“…In particular, if the probability distribution of s is non-vanishing (i.e. all shifts are allowed; S = Z L ) and v i is aperiodic for some i (see [1]), then Σ y is full-rank.…”

Section: The Settingmentioning

confidence: 99%

“…Particularly, much focus was put on the required sample complexity, namely the number of samples N required to achieve a prescribed estimation error for a given SNR value. We also mention [1], where it was shown that estimating the second moment of the measurements y in MRA is sufficient to recover the signal x if the distribution of the shifts is aperiodic, resulting in an improved sample complexity rate. Aside from approaches leveraging shiftinvariant moments, another widespread approach for solving estimation problems akin to MRA (or more generally -estimation problems involving nuisance parameters) is Expectation Maximization (EM) [16].…”

Section: Related Workmentioning

confidence: 99%

“…. , L − 1, where η ∼ CN (0, σ 2 I L ), {v i } r i=1 are orthonormal (since F is unitary), and s is the index of the cyclic shift from (1). Correspondingly, the covariance matrix of x is given by…”

Section: Practical Estimation Procedures With Theoretical Guaranteesmentioning

confidence: 99%

“…(2) ŷ and M (4) ŷ are given by P y and T y , respectively. Since P y and T y are invariant to cyclic shifts in y, and as y = R s {x} (see (1) with σ = 0), P y and T y can be viewed as computed directly from M (2) x and M (4)…”

Section: Practical Estimation Procedures With Theoretical Guaranteesmentioning

confidence: 99%

See 2 more Smart Citations

Multi-reference factor analysis: low-rank covariance estimation under unknown translations

Landa,

Shkolnisky

2019

Preprint

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We consider the problem of estimating the covariance matrix of a random signal observed through unknown translations (modeled by cyclic shifts) and corrupted by noise. Solving this problem allows to discover low-rank structures masked by the existence of translations (which act as nuisance parameters), with direct application to Principal Components Analysis (PCA). We assume that the underlying signal is of length L and follows a standard factor model with mean zero and r normally-distributed factors. To recover the covariance matrix in this case, we propose to employ the second-and fourth-order shiftinvariant moments of the signal known as the power spectrum and the trispectrum. We prove that they are sufficient for recovering the covariance matrix (under a certain technical condition) when r < √ L. Correspondingly, we provide a polynomial-time procedure for estimating the covariance matrix from many (translated and noisy) observations, where no explicit knowledge of r is required, and prove the procedure's statistical consistency. While our results establish that covariance estimation is possible from the power spectrum and the trispectrum for low-rank covariance matrices, we prove that this is not the case for full-rank covariance matrices. We conduct numerical experiments that corroborate our theoretical findings, and demonstrate the favorable performance of our algorithms in various settings, including in high levels of noise.

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Multi-Segment Reconstruction Using Invariant Features

Zehni

Minh

Zhao

2018

2018 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)

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Multi-segment reconstruction (MSR) problem consists of recovering a signal from noisy segments with unknown positions of the observation windows. One example arises in DNA sequence assembly, which is typically solved by matching short reads to form longer sequences. Instead of trying to locate the segment within the sequence through pair-wise matching, we propose a new approach that uses shift-invariant features to estimate both the underlying signal and the distribution of the positions of the segments. Using the invariant features, we formulate the problem as a constrained nonlinear least-squares. The non-convexity of the problem leads to its sensitivity to the initialization. However, with clean data, we show empirically that for longer segment lengths, random initialization achieves exact recovery. Furthermore, we compare the performance of our approach to the results of expectation maximization and demonstrate that the new approach is robust to noise and computationally more efficient.

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Multireference Alignment is Easier with an Aperiodic Translation Distribution

Cited by 6 publications

References 0 publications

Toward single particle reconstruction without particle picking: Breaking the detection limit

Toward single particle reconstruction without particle picking: Breaking the detection limit

Multi-reference factor analysis: low-rank covariance estimation under unknown translations

Multi-Segment Reconstruction Using Invariant Features

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