arXiv subjects

Amit Singer

Publications and source records attributed to Amit Singer.

At least 73 records · Page 4Linked to original sources

arXiv2018

3D ab initio modeling in cryo-EM by autocorrelation analysis

Single-Particle Reconstruction (SPR) in Cryo-Electron Microscopy (cryo-EM) is the task of estimating the 3D structure of a molecule from a set of noisy 2D projections, taken from unknown viewing directions. Many algorithms for SPR start from an initial reference molecule, and alternate between refining the estimated viewing angles given the molecule, and refining the molecule given the viewing angles. This scheme is called iterative refinement. Reliance on an initial, user-chosen reference introduces model bias, and poor initialization can lead to slow convergence. Furthermore, since no ground truth is available for an unsolved molecule, it is difficult to validate the obtained results. This creates the need for high quality ab initio models that can be quickly obtained from experimental data with minimal priors, and which can also be used for validation. We propose a procedure to obtain such an ab initio model directly from raw data using Kam's autocorrelation method. Kam's method has been known since 1980, but it leads to an underdetermined system, with missing orthogonal matrices. Until now, this system has been solved only for special cases, such as highly symmetric molecules or molecules for which a homologous structure was already available. In this paper, we show that knowledge of just two clean projections is sufficient to guarantee a unique solution to the system. This system is solved by an optimization-based heuristic. For the first time, we are then able to obtain a low-resolution ab initio model of an asymmetric molecule directly from raw data, without 2D class averaging and without tilting. Numerical results are presented on both synthetic and experimental data.

math.OC↗

arXiv2017

Synchronization over Cartan motion groups via contraction

Group contraction is an algebraic map that relates two classes of Lie groups by a limiting process. We utilize this notion for the compactification of the class of Cartan motion groups. The compactification process is then applied to reduce a non-compact synchronization problem to a problem where the solution can be obtained by means of a unitary, faithful representation. We describe this method of synchronization via contraction in detail and analyze several important aspects of this application. One important special case of Cartan motion groups is the group of rigid motions, also called the special Euclidean group. We thoroughly discuss the synchronization over this group and show numerically the advantages of our approach compared to some current state-of-the-art synchronization methods on both synthetic and real data.

math.OC↗

arXiv2017

Bispectrum Inversion with Application to Multireference Alignment

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 non-convex optimization over the smooth manifold of phases. Empirically, in the absence of noise, these non-convex 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.

cs.IT↗

arXiv2017

A Survey of Structure from Motion

The structure from motion (SfM) problem in computer vision is the problem of recovering the three-dimensional ($3$D) structure of a stationary scene from a set of projective measurements, represented as a collection of two-dimensional ($2$D) images, via estimation of motion of the cameras corresponding to these images. In essence, SfM involves the three main stages of (1) extraction of features in images (e.g., points of interest, lines, etc.) and matching these features between images, (2) camera motion estimation (e.g., using relative pairwise camera positions estimated from the extracted features), and (3) recovery of the $3$D structure using the estimated motion and features (e.g., by minimizing the so-called reprojection error). This survey mainly focuses on relatively recent developments in the literature pertaining to stages (2) and (3). More specifically, after touching upon the early factorization-based techniques for motion and structure estimation, we provide a detailed account of some of the recent camera location estimation methods in the literature, followed by discussion of notable techniques for $3$D structure recovery. We also cover the basics of the simultaneous localization and mapping (SLAM) problem, which can be viewed as a specific case of the SfM problem. Further, our survey includes a review of the fundamentals of feature extraction and matching (i.e., stage (1) above), various recent methods for handling ambiguities in $3$D scenes, SfM techniques involving relatively uncommon camera models and image features, and popular sources of data and SfM software.

cs.CV↗

arXiv2017

Factor Analysis for Spectral Estimation

Power spectrum estimation is an important tool in many applications, such as the whitening of noise. The popular multitaper method enjoys significant success, but fails for short signals with few samples. We propose a statistical model where a signal is given by a random linear combination of fixed, yet unknown, stochastic sources. Given multiple such signals, we estimate the subspace spanned by the power spectra of these fixed sources. Projecting individual power spectrum estimates onto this subspace increases estimation accuracy. We provide accuracy guarantees for this method and demonstrate it on simulated and experimental data from cryo-electron microscopy.

math.ST↗

arXiv2017

Anisotropic twicing for single particle reconstruction using autocorrelation analysis

The missing phase problem in X-ray crystallography is commonly solved using the technique of molecular replacement, which borrows phases from a previously solved homologous structure, and appends them to the measured Fourier magnitudes of the diffraction patterns of the unknown structure. More recently, molecular replacement has been proposed for solving the missing orthogonal matrices problem arising in Kam's autocorrelation analysis for single particle reconstruction using X-ray free electron lasers and cryo-EM. In classical molecular replacement, it is common to estimate the magnitudes of the unknown structure as twice the measured magnitudes minus the magnitudes of the homologous structure, a procedure known as `twicing'. Mathematically, this is equivalent to finding an unbiased estimator for a complex-valued scalar. We generalize this scheme for the case of estimating real or complex valued matrices arising in single particle autocorrelation analysis. We name this approach "Anisotropic Twicing" because unlike the scalar case, the unbiased estimator is not obtained by a simple magnitude isotropic correction. We compare the performance of the least squares, twicing and anisotropic twicing estimators on synthetic and experimental datasets. We demonstrate 3D homology modeling in cryo-EM directly from experimental data without iterative refinement or class averaging, for the first time.

cs.CV↗

arXiv2017

Continuously heterogeneous hyper-objects in cryo-EM and 3-D movies of many temporal dimensions

Single particle cryo-electron microscopy (EM) is an increasingly popular method for determining the 3-D structure of macromolecules from noisy 2-D images of single macromolecules whose orientations and positions are random and unknown. One of the great opportunities in cryo-EM is to recover the structure of macromolecules in heterogeneous samples, where multiple types or multiple conformations are mixed together. Indeed, in recent years, many tools have been introduced for the analysis of multiple discrete classes of molecules mixed together in a cryo-EM experiment. However, many interesting structures have a continuum of conformations which do not fit discrete models nicely; the analysis of such continuously heterogeneous models has remained a more elusive goal. In this manuscript, we propose to represent heterogeneous molecules and similar structures as higher dimensional objects. We generalize the basic operations used in many existing reconstruction algorithms, making our approach generic in the sense that, in principle, existing algorithms can be adapted to reconstruct those higher dimensional objects. As proof of concept, we present a prototype of a new algorithm which we use to solve simulated reconstruction problems.

cs.CV↗

arXiv2017

Semidefinite Programming Approach for the Quadratic Assignment Problem with a Sparse Graph

The matching problem between two adjacency matrices can be formulated as the NP-hard quadratic assignment problem (QAP). Previous work on semidefinite programming (SDP) relaxations to the QAP have produced solutions that are often tight in practice, but such SDPs typically scale badly, involving matrix variables of dimension $n^2$ where n is the number of nodes. To achieve a speed up, we propose a further relaxation of the SDP involving a number of positive semidefinite matrices of dimension $\mathcal{O}(n)$ no greater than the number of edges in one of the graphs. The relaxation can be further strengthened by considering cliques in the graph, instead of edges. The dual problem of this novel relaxation has a natural three-block structure that can be solved via a convergent Augmented Direction Method of Multipliers (ADMM) in a distributed manner, where the most expensive step per iteration is computing the eigendecomposition of matrices of dimension $\mathcal{O}(n)$. The new SDP relaxation produces strong bounds on quadratic assignment problems where one of the graphs is sparse with reduced computational complexity and running times, and can be used in the context of nuclear magnetic resonance spectroscopy (NMR) to tackle the assignment problem.

math.OC↗

arXiv2017

$e$PCA: High Dimensional Exponential Family PCA

Many applications, such as photon-limited imaging and genomics, involve large datasets with noisy entries from exponential family distributions. It is of interest to estimate the covariance structure and principal components of the noiseless distribution. Principal Component Analysis (PCA), the standard method for this setting, can be inefficient when the noise is non-Gaussian. We develop $e$PCA (exponential family PCA), a new methodology for PCA on exponential family distributions. $e$PCA can be used for dimensionality reduction and denoising of large data matrices. $e$PCA involves the eigendecomposition of a new covariance matrix estimator, constructed in a simple and deterministic way using moment calculations, shrinkage, and random matrix theory. We provide several theoretical justifications for our estimator, including the finite-sample convergence rate, and the Marchenko-Pastur law in high dimensions. $e$PCA compares favorably to PCA and various PCA alternatives for exponential families, in simulations as well as in XFEL and SNP data analysis. An open-source implementation is available.

stat.ME↗

arXiv2017

Disentangling Orthogonal Matrices

Motivated by a certain molecular reconstruction methodology in cryo-electron microscopy, we consider the problem of solving a linear system with two unknown orthogonal matrices, which is a generalization of the well-known orthogonal Procrustes problem. We propose an algorithm based on a semi-definite programming (SDP) relaxation, and give a theoretical guarantee for its performance. Both theoretically and empirically, the proposed algorithm performs better than the naïve approach of solving the linear system directly without the orthogonal constraints. We also consider the generalization to linear systems with more than two unknown orthogonal matrices.

math.OC↗

arXiv2017

Integrating NOE and RDC using sum-of-squares relaxation for protein structure determination

We revisit the problem of protein structure determination from geometrical restraints from NMR, using convex optimization. It is well-known that the NP-hard distance geometry problem of determining atomic positions from pairwise distance restraints can be relaxed into a convex semidefinite program. Often the NOE distance restraints are too imprecise and sparse for accurate structure determination. Residual dipolar coupling (RDC) measurements provide additional geometric information on the angles between atom-pair directions and axes of the principal-axis-frame. The optimization problem involving RDC is highly non-convex and requires a good initialization even within the simulated annealing framework. In this paper, we model the protein backbone as an articulated structure composed of rigid units. Determining the rotation of each rigid unit gives the full protein structure. We propose solving the non-convex optimization problems using the sum-of-squares (SOS) hierarchy. The two algorithms - RDC-SOS and RDC-NOE-SOS, have polynomial time complexity in the number of amino-acid residues and run efficiently on a standard desktop. In many instances, the proposed methods exactly recover the solution to the original non-convex optimization problem. We introduce a statistical tool, the Cramer-Rao bound (CRB), to provide an information theoretic bound on the highest resolution one can hope to achieve when determining protein structure from noisy measurements using any methodology. Our simulation results show that when the RDC measurements are corrupted by Gaussian noise of realistic variance, both SOS based algorithms attain the CRB. We successfully apply our method in a divide-and-conquer fashion to determine the structure of ubiquitin from experimental NOE and RDC measurements, achieving more accurate and faster reconstructions compared to the current state of the art.

cs.CE↗

arXiv2017

A New Rank Constraint on Multi-view Fundamental Matrices, and its Application to Camera Location Recovery

Accurate estimation of camera matrices is an important step in structure from motion algorithms. In this paper we introduce a novel rank constraint on collections of fundamental matrices in multi-view settings. We show that in general, with the selection of proper scale factors, a matrix formed by stacking fundamental matrices between pairs of images has rank 6. Moreover, this matrix forms the symmetric part of a rank 3 matrix whose factors relate directly to the corresponding camera matrices. We use this new characterization to produce better estimations of fundamental matrices by optimizing an L1-cost function using Iterative Re-weighted Least Squares and Alternate Direction Method of Multiplier. We further show that this procedure can improve the recovery of camera locations, particularly in multi-view settings in which fewer images are available.

cs.CV↗

arXiv2017

Sample Complexity of the Boolean Multireference Alignment Problem

The Boolean multireference alignment problem consists in recovering a Boolean signal from multiple shifted and noisy observations. In this paper we obtain an expression for the error exponent of the maximum A posteriori decoder. This expression is used to characterize the number of measurements needed for signal recovery in the low SNR regime, in terms of higher order autocorrelations of the signal. The characterization is explicit for various signal dimensions, such as prime and even dimensions.

cs.IT↗

arXiv2017

Mahalanobis Distance for Class Averaging of Cryo-EM Images

Single particle reconstruction (SPR) from cryo-electron microscopy (EM) is a technique in which the 3D structure of a molecule needs to be determined from its contrast transfer function (CTF) affected, noisy 2D projection images taken at unknown viewing directions. One of the main challenges in cryo-EM is the typically low signal to noise ratio (SNR) of the acquired images. 2D classification of images, followed by class averaging, improves the SNR of the resulting averages, and is used for selecting particles from micrographs and for inspecting the particle images. We introduce a new affinity measure, akin to the Mahalanobis distance, to compare cryo-EM images belonging to different defocus groups. The new similarity measure is employed to detect similar images, thereby leading to an improved algorithm for class averaging. We evaluate the performance of the proposed class averaging procedure on synthetic datasets, obtaining state of the art classification.

stat.AP↗

arXiv2016

A Representation Theory Perspective on Simultaneous Alignment and Classification

One of the difficulties in 3D reconstruction of molecules from images in single particle Cryo-Electron Microscopy (Cryo-EM), in addition to high levels of noise and unknown image orientations, is heterogeneity in samples: in many cases, the samples contain a mixture of molecules, or multiple conformations of one molecule. Many algorithms for the reconstruction of molecules from images in heterogeneous Cryo-EM experiments are based on iterative approximations of the molecules in a non-convex optimization that is prone to reaching suboptimal local minima. Other algorithms require an alignment in order to perform classification, or vice versa. The recently introduced Non-Unique Games framework provides a representation theoretic approach to studying problems of alignment over compact groups, and offers convex relaxations for alignment problems which are formulated as semidefinite programs (SDPs) with certificates of global optimality under certain circumstances. In this manuscript, we propose to extend Non-Unique Games to the problem of simultaneous alignment and classification with the goal of simultaneously classifying Cryo-EM images and aligning them within their respective classes. Our proposed approach can also be extended to the case of continuous heterogeneity.

cs.CV↗

arXiv2016

Bias Correction in Saupe Tensor Estimation

Estimation of the Saupe tensor is central to the determination of molecular structures from residual dipolar couplings (RDC) or chemical shift anisotropies. Assuming a given template structure, the singular value decomposition (SVD) method proposed in Losonczi et al. 1999 has been used traditionally to estimate the Saupe tensor. Despite its simplicity, whenever the template structure has large structural noise, the eigenvalues of the estimated tensor have a magnitude systematically smaller than their actual values. This leads to systematic error when calculating the eigenvalue dependent parameters, magnitude and rhombicity. We propose here a Monte Carlo simulation method to remove such bias. We further demonstrate the effectiveness of our method in the setting when the eigenvalue estimates from multiple template protein fragments are available and their average is used as an improved eigenvalue estimator. For both synthetic and experimental RDC datasets of ubiquitin, when using template fragments corrupted by large noise, the magnitude of our proposed bias-reduced estimator generally reaches at least 90% of the actual value, whereas the magnitude of SVD estimator can be shrunk below 80% of the true value.

cs.CE↗

arXiv2016

Tightness of the maximum likelihood semidefinite relaxation for angular synchronization

Maximum likelihood estimation problems are, in general, intractable optimization problems. As a result, it is common to approximate the maximum likelihood estimator (MLE) using convex relaxations. In some cases, the relaxation is tight: it recovers the true MLE. Most tightness proofs only apply to situations where the MLE exactly recovers a planted solution (known to the analyst). It is then sufficient to establish that the optimality conditions hold at the planted signal. In this paper, we study an estimation problem (angular synchronization) for which the MLE is not a simple function of the planted solution, yet for which the convex relaxation is tight. To establish tightness in this context, the proof is less direct because the point at which to verify optimality conditions is not known explicitly. Angular synchronization consists in estimating a collection of $n$ phases, given noisy measurements of the pairwise relative phases. The MLE for angular synchronization is the solution of a (hard) non-bipartite Grothendieck problem over the complex numbers. We consider a stochastic model for the data: a planted signal (that is, a ground truth set of phases) is corrupted with non-adversarial random noise. Even though the MLE does not coincide with the planted signal, we show that the classical semidefinite relaxation for it is tight, with high probability. This holds even for high levels of noise.

math.OC↗

arXiv2016

Denoising and Covariance Estimation of Single Particle Cryo-EM Images

The problem of image restoration in cryo-EM entails correcting for the effects of the Contrast Transfer Function (CTF) and noise. Popular methods for image restoration include `phase flipping', which corrects only for the Fourier phases but not amplitudes, and Wiener filtering, which requires the spectral signal to noise ratio. We propose a new image restoration method which we call `Covariance Wiener Filtering' (CWF). In CWF, the covariance matrix of the projection images is used within the classical Wiener filtering framework for solving the image restoration deconvolution problem. Our estimation procedure for the covariance matrix is new and successfully corrects for the CTF. We demonstrate the efficacy of CWF by applying it to restore both simulated and experimental cryo-EM images. Results with experimental datasets demonstrate that CWF provides a good way to evaluate the particle images and to see what the dataset contains even without 2D classification and averaging.

cs.CV↗