arXiv subjects

Amit Singer

Publications and source records attributed to Amit Singer.

At least 37 records · Page 2Linked to original sources

arXiv2022

Multi-target detection with rotations

We consider the multi-target detection problem of estimating a two-dimensional target image from a large noisy measurement image that contains many randomly rotated and translated copies of the target image. Motivated by single-particle cryo-electron microscopy, we focus on the low signal-to-noise regime, where it is difficult to estimate the locations and orientations of the target images in the measurement. Our approach uses autocorrelation analysis to estimate rotationally and translationally invariant features of the target image. We demonstrate that, regardless of the level of noise, our technique can be used to recover the target image when the measurement is sufficiently large.

eess.IV↗

arXiv2022

Quantitatively visualizing bipartite datasets

As experiments continue to increase in size and scope, a fundamental challenge of subsequent analyses is to recast the wealth of information into an intuitive and readily-interpretable form. Often, each measurement only conveys the relationship between a pair of entries, and it is difficult to integrate these local interactions across a dataset to form a cohesive global picture. The classic localization problem tackles this question, transforming local measurements into a global map that reveals the underlying structure of a system. Here, we examine the more challenging bipartite localization problem, where pairwise distances are only available for bipartite data comprising two classes of entries (such as antibody-virus interactions, drug-cell potency, or user-rating profiles). We modify previous algorithms to solve bipartite localization and examine how each method behaves in the presence of noise, outliers, and partially-observed data. As a proof of concept, we apply these algorithms to antibody-virus neutralization measurements to create a basis set of antibody behaviors, formalize how potently inhibiting some viruses necessitates weakly inhibiting other viruses, and quantify how often combinations of antibodies exhibit degenerate behavior.

physics.bio-ph↗

arXiv2022

Ab-initio Contrast Estimation and Denoising of Cryo-EM Images

Background and Objective: The contrast of cryo-EM images varies from one to another, primarily due to the uneven thickness of the ice layer. This contrast variation can affect the quality of 2-D class averaging, 3-D ab-initio modeling, and 3-D heterogeneity analysis. Contrast estimation is currently performed during 3-D iterative refinement. As a result, the estimates are not available at the earlier computational stages of class averaging and ab-initio modeling. This paper aims to solve the contrast estimation problem directly from the picked particle images in the ab-initio stage, without estimating the 3-D volume, image rotations, or class averages. Methods: The key observation underlying our analysis is that the 2-D covariance matrix of the raw images is related to the covariance of the underlying clean images, the noise variance, and the contrast variability between images. We show that the contrast variability can be derived from the 2-D covariance matrix and we apply the existing Covariance Wiener Filtering (CWF) framework to estimate it. We also demonstrate a modification of CWF to estimate the contrast of individual images. Results: Our method improves the contrast estimation by a large margin, compared to the previous CWF method. Its estimation accuracy is often comparable to that of an oracle that knows the ground truth covariance of the clean images. The more accurate contrast estimation also improves the quality of image restoration as demonstrated in both synthetic and experimental datasets. Conclusions: This paper proposes an effective method for contrast estimation directly from noisy images without using any 3-D volume information. It enables contrast correction in the earlier stage of single particle analysis, and may improve the accuracy of downstream processing.

cs.CV↗

arXiv2022

A Molecular Prior Distribution for Bayesian Inference Based on Wilson Statistics

Background and Objective: Wilson statistics describe well the power spectrum of proteins at high frequencies. Therefore, it has found several applications in structural biology, e.g., it is the basis for sharpening steps used in cryogenic electron microscopy (cryo-EM). A recent paper gave the first rigorous proof of Wilson statistics based on a formalism of Wilson's original argument. This new analysis also leads to statistical estimates of the scattering potential of proteins that reveal a correlation between neighboring Fourier coefficients. Here we exploit these estimates to craft a novel prior that can be used for Bayesian inference of molecular structures. Methods: We describe the properties of the prior and the computation of its hyperparameters. We then evaluate the prior on two synthetic linear inverse problems, and compare against a popular prior in cryo-EM reconstruction at a range of SNRs. Results: We show that the new prior effectively suppresses noise and fills-in low SNR regions in the spectral domain. Furthermore, it improves the resolution of estimates on the problems considered for a wide range of SNR and produces Fourier Shell Correlation curves that are insensitive to masking effects. Conclusions: We analyze the assumptions in the model, discuss relations to other regularization strategies, and postulate on potential implications for structure determination in cryo-EM.

q-bio.QM↗

arXiv2022

An approximate expectation-maximization for two-dimensional multi-target detection

We consider the two-dimensional multi-target detection (MTD) problem of estimating a target image from a noisy measurement that contains multiple copies of the image, each randomly rotated and translated. The MTD model serves as a mathematical abstraction of the structure reconstruction problem in single-particle cryo-electron microscopy, the chief motivation of this study. We focus on high noise regimes, where accurate detection of image occurrences within a measurement is impossible. To estimate the image, we develop an expectation-maximization framework that aims to maximize an approximation of the likelihood function. We demonstrate image recovery in highly noisy environments, and show that our framework outperforms the previously studied autocorrelation analysis in a wide range of parameters.

eess.SP↗

arXiv2022

On a linearization of quadratic Wasserstein distance

This paper studies the problem of computing a linear approximation of quadratic Wasserstein distance $W_2$. In particular, we compute an approximation of the negative homogeneous weighted Sobolev norm whose connection to Wasserstein distance follows from a classic linearization of a general Monge-Ampére equation. Our contribution is threefold. First, we provide expository material on this classic linearization of Wasserstein distance including a quantitative error estimate. Second, we reduce the computational problem to solving an elliptic boundary value problem involving the Witten Laplacian, which is a Schrödinger operator of the form $H = -Δ+ V$, and describe an associated embedding. Third, for the case of probability distributions on the unit square $[0,1]^2$ represented by $n \times n$ arrays we present a fast code demonstrating our approach. Several numerical examples are presented.

math.NA↗

arXiv2021

Sparse multi-reference alignment: sample complexity and computational hardness

Motivated by the problem of determining the atomic structure of macromolecules using single-particle cryo-electron microscopy (cryo-EM), we study the sample and computational complexities of the sparse multi-reference alignment (MRA) model: the problem of estimating a sparse signal from its noisy, circularly shifted copies. Based on its tight connection to the crystallographic phase retrieval problem, we establish that if the number of observations is proportional to the square of the variance of the noise, then the sparse MRA problem is statistically feasible for sufficiently sparse signals. To investigate its computational hardness, we consider three types of computational frameworks: projection-based algorithms, bispectrum inversion, and convex relaxations. We show that a state-of-the-art projection-based algorithm achieves the optimal estimation rate, but its computational complexity is exponential in the sparsity level. The bispectrum framework provides a statistical-computational trade-off: it requires more observations (so its estimation rate is suboptimal), but its computational load is provably polynomial in the signal's length. The convex relaxation approach provides polynomial time algorithms (with a large exponent) that recover sufficiently sparse signals at the optimal estimation rate. We conclude the paper by discussing potential statistical and algorithmic implications for cryo-EM.

cs.IT↗

arXiv2021

NMR Assignment through Linear Programming

Nuclear Magnetic Resonance (NMR) Spectroscopy is the second most used technique (after X-ray crystallography) for structural determination of proteins. A computational challenge in this technique involves solving a discrete optimization problem that assigns the resonance frequency to each atom in the protein. This paper introduces LIAN (LInear programming Assignment for NMR), a novel linear programming formulation of the problem which yields state-of-the-art results in simulated and experimental datasets.

cs.AI↗

arXiv2021

Manifold learning with arbitrary norms

Manifold learning methods play a prominent role in nonlinear dimensionality reduction and other tasks involving high-dimensional data sets with low intrinsic dimensionality. Many of these methods are graph-based: they associate a vertex with each data point and a weighted edge with each pair. Existing theory shows that the Laplacian matrix of the graph converges to the Laplace-Beltrami operator of the data manifold, under the assumption that the pairwise affinities are based on the Euclidean norm. In this paper, we determine the limiting differential operator for graph Laplacians constructed using $\textit{any}$ norm. Our proof involves an interplay between the second fundamental form of the manifold and the convex geometry of the given norm's unit ball. To demonstrate the potential benefits of non-Euclidean norms in manifold learning, we consider the task of mapping the motion of large molecules with continuous variability. In a numerical simulation we show that a modified Laplacian eigenmaps algorithm, based on the Earthmover's distance, outperforms the classic Euclidean Laplacian eigenmaps, both in terms of computational cost and the sample size needed to recover the intrinsic geometry.

cs.LG↗

arXiv2021

The generalized orthogonal Procrustes problem in the high noise regime

We consider the problem of estimating a cloud of points from numerous noisy observations of that cloud after unknown rotations, and possibly reflections. This is an instance of the general problem of estimation under group action, originally inspired by applications in 3-D imaging and computer vision. We focus on a regime where the noise level is larger than the magnitude of the signal, so much so that the rotations cannot be estimated reliably. We propose a simple and efficient procedure based on invariant polynomials (effectively: the Gram matrices) to recover the signal, and we assess it against fundamental limits of the problem that we derive. We show our approach adapts to the noise level and is statistically optimal (up to constants) for both the low and high noise regimes. In studying the variance of our estimator, we encounter the question of the sensivity of a type of thin Cholesky factorization, for which we provide an improved bound which may be of independent interest.

math.ST↗

arXiv2021

Centering noisy images with application to cryo-EM

We target the problem of estimating the center of mass of noisy 2-D images. We assume that the noise dominates the image, and thus many standard approaches are vulnerable to estimation errors. Our approach uses a surrogate function to the geometric median, which is a robust estimator of the center of mass. We mathematically analyze cases in which the geometric median fails to provide a reasonable estimate of the center of mass, and prove that our surrogate function leads to a successful estimate. One particular application for our method is to improve 3-D reconstruction in single-particle cryo-electron microscopy (cryo-EM). We show how to apply our approach for a better translational alignment of macromolecules picked from experimental data. In this way, we facilitate the succeeding steps of reconstruction and streamline the entire cryo-EM pipeline, saving valuable computational time and supporting resolution enhancement.

eess.IV↗

arXiv2021

Random Conical Tilt Reconstruction without Particle Picking in Cryo-electron Microscopy

We propose a method to reconstruct the 3-D molecular structure from micrographs collected at just one sample tilt angle in the random conical tilt scheme in cryo-electron microscopy. Our method uses autocorrelation analysis on the micrographs to estimate features of the molecule which are invariant under certain nuisance parameters such as the positions of molecular projections in the micrographs. This enables us to reconstruct the molecular structure directly from micrographs, completely circumventing the need for particle picking. We demonstrate reconstructions with simulated data and investigate the effect of the missing-cone region. These results show promise to reduce the size limit for single particle reconstruction in cryo-electron microscopy.

physics.data-an↗

arXiv2020

Super-resolution multi-reference alignment

We study super-resolution multi-reference alignment, the problem of estimating a signal from many circularly shifted, down-sampled, and noisy observations. We focus on the low SNR regime, and show that a signal in $\mathbb{R}^M$ is uniquely determined when the number $L$ of samples per observation is of the order of the square root of the signal's length $(L=O(\sqrt{M}))$. Phrased more informally, one can square the resolution. This result holds if the number of observations is proportional to at least 1/SNR$^3$. In contrast, with fewer observations recovery is impossible even when the observations are not down-sampled ($L=M$). The analysis combines tools from statistical signal processing and invariant theory. We design an expectation-maximization algorithm and demonstrate that it can super-resolve the signal in challenging SNR regimes.

cs.IT↗

arXiv2020

Wasserstein K-Means for Clustering Tomographic Projections

Motivated by the 2D class averaging problem in single-particle cryo-electron microscopy (cryo-EM), we present a k-means algorithm based on a rotationally-invariant Wasserstein metric for images. Unlike existing methods that are based on Euclidean ($L_2$) distances, we prove that the Wasserstein metric better accommodates for the out-of-plane angular differences between different particle views. We demonstrate on a synthetic dataset that our method gives superior results compared to an $L_2$ baseline. Furthermore, there is little computational overhead, thanks to the use of a fast linear-time approximation to the Wasserstein-1 metric, also known as the Earthmover's distance.

cs.CV↗

arXiv2020

Product Manifold Learning

We consider problems of dimensionality reduction and learning data representations for continuous spaces with two or more independent degrees of freedom. Such problems occur, for example, when observing shapes with several components that move independently. Mathematically, if the parameter space of each continuous independent motion is a manifold, then their combination is known as a product manifold. In this paper, we present a new paradigm for non-linear independent component analysis called manifold factorization. Our factorization algorithm is based on spectral graph methods for manifold learning and the separability of the Laplacian operator on product spaces. Recovering the factors of a manifold yields meaningful lower-dimensional representations and provides a new way to focus on particular aspects of the data space while ignoring others. We demonstrate the potential use of our method for an important and challenging problem in structural biology: mapping the motions of proteins and other large molecules using cryo-electron microscopy datasets.

cs.LG↗

arXiv2020

Computational Methods for Single-Particle Cryo-EM

Single-particle electron cryomicroscopy (cryo-EM) is an increasingly popular technique for elucidating the three-dimensional structure of proteins and other biologically significant complexes at near-atomic resolution. It is an imaging method that does not require crystallization and can capture molecules in their native states. In single-particle cryo-EM, the three-dimensional molecular structure needs to be determined from many noisy two-dimensional tomographic projections of individual molecules, whose orientations and positions are unknown. The high level of noise and the unknown pose parameters are two key elements that make reconstruction a challenging computational problem. Even more challenging is the inference of structural variability and flexible motions when the individual molecules being imaged are in different conformational states. This review discusses computational methods for structure determination by single-particle cryo-EM and their guiding principles from statistical inference, machine learning, and signal processing that also play a significant role in many other data science applications.

physics.comp-ph↗

arXiv2020

Bias and variance reduction and denoising for CTF Estimation

When using an electron microscope for imaging of particles embedded in vitreous ice, the objective lens will inevitably corrupt the projection images. This corruption manifests as a band-pass filter on the micrograph. In addition, it causes the phase of several frequency bands to be flipped and distorts frequency bands. As a precursor to compensating for this distortion, the corrupting point spread function, which is termed the contrast transfer function (CTF) in reciprocal space, must be estimated. In this paper, we will present a novel method for CTF estimation. Our method is based on the multi-taper method for power spectral density estimation, which aims to reduce the bias and variance of the estimator. Furthermore, we use known properties of the CTF and of the background of the power spectrum to increase the accuracy of our estimation. We will show that the resulting estimates capture the zero-crossings of the CTF in the low-mid frequency range.

eess.IV↗

arXiv2020

Multi-target Detection with an Arbitrary Spacing Distribution

Motivated by the structure reconstruction problem in single-particle cryo-electron microscopy, we consider the multi-target detection model, where multiple copies of a target signal occur at unknown locations in a long measurement, further corrupted by additive Gaussian noise. At low noise levels, one can easily detect the signal occurrences and estimate the signal by averaging. However, in the presence of high noise, which is the focus of this paper, detection is impossible. Here, we propose two approaches---autocorrelation analysis and an approximate expectation maximization algorithm---to reconstruct the signal without the need to detect signal occurrences in the measurement. In particular, our methods apply to an arbitrary spacing distribution of signal occurrences. We demonstrate reconstructions with synthetic data and empirically show that the sample complexity of both methods scales as 1/SNR^3 in the low SNR regime.

cs.IT↗