Lederman, Roy R; Andén, Joakim; Singer, Amit Hyper-molecules: on the representation and recovery of dynamical structures for applications in flexible macro-molecules in cryo-EM Journal Article Inverse Problems, 36 (4), pp. 044005, 2020, ISSN: 0266-5611, 1361-6420. Links | BibTeX | Tags: cryo-EM, heterogeneity, HyperMolecules, MCMC, Variational inference @article{lederman_hyper-molecules_2020, title = {Hyper-molecules: on the representation and recovery of dynamical structures for applications in flexible macro-molecules in cryo-EM}, author = {Roy R Lederman and Joakim Andén and Amit Singer}, url = {https://iopscience.iop.org/article/10.1088/1361-6420/ab5ede}, doi = {10.1088/1361-6420/ab5ede}, issn = {0266-5611, 1361-6420}, year = {2020}, date = {2020-04-01}, urldate = {2020-08-13}, journal = {Inverse Problems}, volume = {36}, number = {4}, pages = {044005}, keywords = {cryo-EM, heterogeneity, HyperMolecules, MCMC, Variational inference}, pubstate = {published}, tppubtype = {article} } |

Lederman, Roy R; Singer, Amit A representation theory perspective on simultaneous alignment and classification Journal Article Applied and Computational Harmonic Analysis, 49 (3), pp. 1001–1024, 2020, ISSN: 1063-5203. Abstract | Links | BibTeX | Tags: Algorithms, Alignment, Classification, cryo-EM, Graph-cut, heterogeneity, Heterogeneous multireference alignment, Representation Theory, Rotation group, SDP, Synchronization @article{lederman_representation_2020, title = {A representation theory perspective on simultaneous alignment and classification}, author = {Roy R Lederman and Amit Singer}, url = {http://www.sciencedirect.com/science/article/pii/S1063520319301034}, doi = {10.1016/j.acha.2019.05.005}, issn = {1063-5203}, year = {2020}, date = {2020-01-01}, urldate = {2021-01-22}, journal = {Applied and Computational Harmonic Analysis}, volume = {49}, number = {3}, pages = {1001--1024}, abstract = {Single particle cryo-electron microscopy (EM) is a method for determining the 3-D structure of macromolecules from many noisy 2-D projection images of individual macromolecules whose orientations and positions are random and unknown. The problem of orientation assignment for the images motivated work on multireference alignment. The recent non-unique games framework provides a representation theoretic approach to alignment over compact groups, and offers a convex relaxation with certificates of global optimality in some cases. One of the great opportunities in cryo-EM is studying heterogeneous samples, containing two or more distinct conformations of molecules. Taking advantage of this opportunity presents an algorithmic challenge: determining both the class and orientation of each particle. We generalize multireference alignment to a problem of alignment and classification, and 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 classes.}, keywords = {Algorithms, Alignment, Classification, cryo-EM, Graph-cut, heterogeneity, Heterogeneous multireference alignment, Representation Theory, Rotation group, SDP, Synchronization}, pubstate = {published}, tppubtype = {article} } Single particle cryo-electron microscopy (EM) is a method for determining the 3-D structure of macromolecules from many noisy 2-D projection images of individual macromolecules whose orientations and positions are random and unknown. The problem of orientation assignment for the images motivated work on multireference alignment. The recent non-unique games framework provides a representation theoretic approach to alignment over compact groups, and offers a convex relaxation with certificates of global optimality in some cases. One of the great opportunities in cryo-EM is studying heterogeneous samples, containing two or more distinct conformations of molecules. Taking advantage of this opportunity presents an algorithmic challenge: determining both the class and orientation of each particle. We generalize multireference alignment to a problem of alignment and classification, and 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 classes. |

Boumal, N; Bendory, T; Lederman, Roy R; Singer, A Heterogeneous multireference alignment: A single pass approach Inproceedings 2018 52nd Annual Conference on Information Sciences and Systems (CISS), pp. 1–6, 2018. Abstract | Links | BibTeX | Tags: bispectrum, concave programming, cryo-EM, cyclic shifts, Discrete Fourier transforms, estimation theory, expectation-maximization, Gaussian mixture models, heterogeneity, heterogeneous MRA, Heterogeneous multireference alignment, Multireference alignment, Noise measurement, non-convex optimization, nonconvex optimization problem, Optimization, Reliability, signal estimation, signal processing, Signal resolution, Signal to noise ratio, single pass approach, Standards @inproceedings{boumal_heterogeneous_2018, title = {Heterogeneous multireference alignment: A single pass approach}, author = {N Boumal and T Bendory and Roy R Lederman and A Singer}, doi = {10.1109/CISS.2018.8362313}, year = {2018}, date = {2018-01-01}, booktitle = {2018 52nd Annual Conference on Information Sciences and Systems (CISS)}, pages = {1--6}, abstract = {Multireference alignment (MRA) is the problem of estimating a signal from many noisy and cyclically shifted copies of itself. In this paper, we consider an extension called heterogeneous MRA, where K signals must be estimated, and each observation comes from one of those signals, unknown to us. This is a simplified model for the heterogeneity problem notably arising in cryo-electron microscopy. We propose an algorithm which estimates the K signals without estimating either the shifts or the classes of the observations. It requires only one pass over the data and is based on low-order moments that are invariant under cyclic shifts. Given sufficiently many measurements, one can estimate these invariant features averaged over the K signals. We then design a smooth, non-convex optimization problem to compute a set of signals which are consistent with the estimated averaged features. We find that, in many cases, the proposed approach estimates the set of signals accurately despite non-convexity, and conjecture the number of signals K that can be resolved as a function of the signal length L is on the order of √L.}, keywords = {bispectrum, concave programming, cryo-EM, cyclic shifts, Discrete Fourier transforms, estimation theory, expectation-maximization, Gaussian mixture models, heterogeneity, heterogeneous MRA, Heterogeneous multireference alignment, Multireference alignment, Noise measurement, non-convex optimization, nonconvex optimization problem, Optimization, Reliability, signal estimation, signal processing, Signal resolution, Signal to noise ratio, single pass approach, Standards}, pubstate = {published}, tppubtype = {inproceedings} } Multireference alignment (MRA) is the problem of estimating a signal from many noisy and cyclically shifted copies of itself. In this paper, we consider an extension called heterogeneous MRA, where K signals must be estimated, and each observation comes from one of those signals, unknown to us. This is a simplified model for the heterogeneity problem notably arising in cryo-electron microscopy. We propose an algorithm which estimates the K signals without estimating either the shifts or the classes of the observations. It requires only one pass over the data and is based on low-order moments that are invariant under cyclic shifts. Given sufficiently many measurements, one can estimate these invariant features averaged over the K signals. We then design a smooth, non-convex optimization problem to compute a set of signals which are consistent with the estimated averaged features. We find that, in many cases, the proposed approach estimates the set of signals accurately despite non-convexity, and conjecture the number of signals K that can be resolved as a function of the signal length L is on the order of √L. |

Lederman, Roy R; Singer, Amit Continuously heterogeneous hyper-objects in cryo-EM and 3-Đ movies of many temporal dimensions Technical Report (arXiv:1704.02899 [cs]), 2017, (arXiv: 1704.02899). Abstract | Links | BibTeX | Tags: Computer Science - Computer Vision and Pattern Recognition, cryo-EM, heterogeneity, HyperMolecules @techreport{lederman_continuously_2017, title = {Continuously heterogeneous hyper-objects in cryo-EM and 3-Đ movies of many temporal dimensions}, author = {Roy R Lederman and Amit Singer}, url = {http://arxiv.org/abs/1704.02899}, year = {2017}, date = {2017-04-01}, urldate = {2020-08-13}, number = {arXiv:1704.02899 [cs]}, abstract = {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.}, note = {arXiv: 1704.02899}, keywords = {Computer Science - Computer Vision and Pattern Recognition, cryo-EM, heterogeneity, HyperMolecules}, pubstate = {published}, tppubtype = {techreport} } 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. |