I co-organize the One World Cryo-EM online seminar series about mathematics and algorithms in cryo-electron microscopy. Please contact me if you are interested in speaking or getting involved in our activities around the seminar.

I am looking for extraordinary postdocs and extraordinary graduate students (at Yale, any graduate program). More information about postdoc positions is available here.

Office hours available here and on Canvas (internal Yale system).

Structural biology and cryo-EM: inverse problems and unsupervised learning, applications of representation theory, numerical analysis, and data organization to imaging of molecules

Computational biology: fast search algorithms, statistics of DNA, sequencing, organization of biological data

CRYO-EM (a little out of date – more updates coming soon)

Cryo-electron microscopy (cryo-EM) is a method for imaging molecules without crystallization. The Nobel Prize in Chemistry 2017 was awarded to Jacques Dubochet, Joachim Frank and Richard Henderson “for the development of cryo-electron microscopy, which both simplifies and improves the imaging of biomolecules.” I work on various problems of alignment, classification and signal processing that are motivated by application in cryo-EM with many other applications. I am particularly interested in heterogeneity, i.e. imaging of mixtures of different types of molecules.

I work on “hyper-molecules” which represent heterogeneous molecules as higher-dimension objects. The movie below is an example of a reconstruction of a continuously heterogeneous object, using the approach described in this paper and this paper.

This is one of several approaches that I am developing for the heterogeneity problem in cryo-EM, and for other aspects of cryo-EM. For more information on my work in cryo-EM, see project page.

The Truncated Fourier Transform and its eigenfunctions, Prolate Spheroidal Wave Functions (PSWF) and Generalized Prolate Spheroidal Functions (GPSF) (also known as Slepian Functions) are frequently encountered in mathematics, physics, signal processing, optics and other areas. Surprisingly, very few resources and code for the numerical computation of GPSFs and their eigenvalues are publicly available. Our sample implementation and associated paper are available at http://github.com/lederman/prol. The code also contains an experimental “open-source proof,” which is code for analytical proofs of some of the results that appear in this paper.

The Laplace Transform and Grunbaum Functions

The Laplace transform is frequently encountered in mathematics, physics, engineering and other areas. However, the spectral properties of the Laplace transform tend to complicate its numerical treatment; therefore, the closely related “Truncated” Laplace Transforms are often used in applications.

Lower bounds on the truncated Fourier transform and truncated Laplace transform: see paper.

GEOMETRY OF DATA

Alternating Diffusion, a method for recovering the common variable in multi-sensor experiments, is discussed in this paper, this technical report and this project webpage. A different approach to the common variable recovery problem, which also constructs representations that are invariable to unknown transformations, is discussed in this paper.

Additional Application: Assembly. The algorithm is also used to construct approximate overlap graphs. These graph are used for fast assembly. Unlike other algorithms, this algorithm allows errors in the reads, so no error-correction is necessary prior to the construction of the graph. See: paper.

@article{herreros_approximating_2021,
title = {Approximating deformation fields for the analysis of continuous heterogeneity of biological macromolecules by 3D Zernike polynomials},
author = {D Herreros and Roy R Lederman and J Krieger and A Jiménez-Moreno and M Martínez and D Myška and D Strelak and J Filipovic and I Bahar and J M Carazo and C O S Sanchez},
url = {https://journals.iucr.org/m/issues/2021/06/00/eh5012/},
doi = {10.1107/S2052252521008903},
issn = {2052-2525},
year = {2021},
date = {2021-11-01},
urldate = {2021-10-26},
journal = {IUCrJ},
volume = {8},
number = {6},
abstract = {A new tool based on 3D Zernike polynomials is presented that allows the study of the continuous heterogeneity of biological macromolecules, revealing the structural relationships present among different states by the approximation of deformation fields.},
note = {Number: 6
Publisher: International Union of Crystallography},
keywords = {cryo-EM, heterogeneity, Zernike},
pubstate = {published},
tppubtype = {article}
}

A new tool based on 3D Zernike polynomials is presented that allows the study of the continuous heterogeneity of biological macromolecules, revealing the structural relationships present among different states by the approximation of deformation fields.

Calero, David Herreros; Lederman, Roy R; Krieger, James; Myška, David; Strelak, David; Filipovic, Jiri; Bahar, Ivet; Carazo, Jose Maria; Sorzano, Carlos Oscar

@article{calero_continuous_2021,
title = {Continuous heterogeneity analysis of CryoEM images through Zernike polynomials and spherical harmonics},
author = {David Herreros Calero and Roy R Lederman and James Krieger and David Myška and David Strelak and Jiri Filipovic and Ivet Bahar and Jose Maria Carazo and Carlos Oscar Sorzano},
url = {https://www.cambridge.org/core/journals/microscopy-and-microanalysis/article/continuous-heterogeneity-analysis-of-cryoem-images-through-zernike-polynomials-and-spherical-harmonics/2A8C58651F413C8A0D66071CB4BC9AAD},
doi = {10.1017/S1431927621006176},
issn = {1431-9276, 1435-8115},
year = {2021},
date = {2021-08-01},
urldate = {2021-08-03},
journal = {Microscopy and Microanalysis},
volume = {27},
number = {S1},
pages = {1680--1682},
abstract = {//static.cambridge.org/content/id/urn%3Acambridge.org%3Aid%3Aarticle%3AS1431927621006176/resource/name/firstPage-S1431927621006176a.jpg},
note = {Publisher: Cambridge University Press},
keywords = {cryo-EM, heterogeneity, Zernike},
pubstate = {published},
tppubtype = {article}
}

@techreport{brofos_manifold_2021,
title = {Manifold Density Estimation via Generalized Dequantization},
author = {James A Brofos and Marcus A Brubaker and Roy R Lederman},
url = {http://arxiv.org/abs/2102.07143},
year = {2021},
date = {2021-07-01},
urldate = {2021-07-14},
abstract = {Density estimation is an important technique for characterizing distributions given observations. Much existing research on density estimation has focused on cases wherein the data lies in a Euclidean space. However, some kinds of data are not well-modeled by supposing that their underlying geometry is Euclidean. Instead, it can be useful to model such data as lying on a textbackslashit manifold with some known structure. For instance, some kinds of data may be known to lie on the surface of a sphere. We study the problem of estimating densities on manifolds. We propose a method, inspired by the literature on "dequantization," which we interpret through the lens of a coordinate transformation of an ambient Euclidean space and a smooth manifold of interest. Using methods from normalizing flows, we apply this method to the dequantization of smooth manifold structures in order to model densities on the sphere, tori, and the orthogonal group.},
note = {arXiv: 2102.07143},
keywords = {Algorithms, Computer Science - Machine Learning, Density estimation, Manifolds, Statistics - Machine Learning},
pubstate = {published},
tppubtype = {techreport}
}

Density estimation is an important technique for characterizing distributions given observations. Much existing research on density estimation has focused on cases wherein the data lies in a Euclidean space. However, some kinds of data are not well-modeled by supposing that their underlying geometry is Euclidean. Instead, it can be useful to model such data as lying on a textbackslashit manifold with some known structure. For instance, some kinds of data may be known to lie on the surface of a sphere. We study the problem of estimating densities on manifolds. We propose a method, inspired by the literature on "dequantization," which we interpret through the lens of a coordinate transformation of an ambient Euclidean space and a smooth manifold of interest. Using methods from normalizing flows, we apply this method to the dequantization of smooth manifold structures in order to model densities on the sphere, tori, and the orthogonal group.

@techreport{fan_maximum_2021,
title = {Maximum likelihood for high-noise group orbit estimation and single-particle cryo-EM},
author = {Zhou Fan and Roy R Lederman and Yi Sun and Tianhao Wang and Sheng Xu},
url = {http://arxiv.org/abs/2107.01305},
year = {2021},
date = {2021-07-01},
urldate = {2021-07-14},
abstract = {Motivated by applications to single-particle cryo-electron microscopy (cryo-EM), we study several problems of function estimation in a low SNR regime, where samples are observed under random rotations of the function domain. In a general framework of group orbit estimation with linear projection, we describe a stratification of the Fisher information eigenvalues according to a sequence of transcendence degrees in the invariant algebra, and relate critical points of the log-likelihood landscape to a sequence of method-of-moments optimization problems. This extends previous results for a discrete rotation group without projection. We then compute these transcendence degrees and the forms of these moment optimization problems for several examples of function estimation under $SO(2)$ and $SO(3)$ rotations, including a simplified model of cryo-EM as introduced by Bandeira, Blum-Smith, Kileel, Perry, Weed, and Wein. For several of these examples, we affirmatively resolve numerical conjectures that $3textasciicircumtextbackslashtextrd$-order moments are sufficient to locally identify a generic signal up to its rotational orbit. For low-dimensional approximations of the electric potential maps of two small protein molecules, we empirically verify that the noise-scalings of the Fisher information eigenvalues conform with these theoretical predictions over a range of SNR, in a model of $SO(3)$ rotations without projection.},
note = {arXiv: 2107.01305},
keywords = {Computer Science - Information Theory, Mathematics - Optimization and Control, Mathematics - Statistics Theory},
pubstate = {published},
tppubtype = {techreport}
}

Motivated by applications to single-particle cryo-electron microscopy (cryo-EM), we study several problems of function estimation in a low SNR regime, where samples are observed under random rotations of the function domain. In a general framework of group orbit estimation with linear projection, we describe a stratification of the Fisher information eigenvalues according to a sequence of transcendence degrees in the invariant algebra, and relate critical points of the log-likelihood landscape to a sequence of method-of-moments optimization problems. This extends previous results for a discrete rotation group without projection. We then compute these transcendence degrees and the forms of these moment optimization problems for several examples of function estimation under $SO(2)$ and $SO(3)$ rotations, including a simplified model of cryo-EM as introduced by Bandeira, Blum-Smith, Kileel, Perry, Weed, and Wein. For several of these examples, we affirmatively resolve numerical conjectures that $3textasciicircumtextbackslashtextrd$-order moments are sufficient to locally identify a generic signal up to its rotational orbit. For low-dimensional approximations of the electric potential maps of two small protein molecules, we empirically verify that the noise-scalings of the Fisher information eigenvalues conform with these theoretical predictions over a range of SNR, in a model of $SO(3)$ rotations without projection.

@techreport{brofos_magnetic_2020,
title = {Magnetic Manifold Hamiltonian Monte Carlo},
author = {James A Brofos and Roy R Lederman},
url = {http://arxiv.org/abs/2010.07753},
year = {2020},
date = {2020-10-01},
urldate = {2020-11-25},
abstract = {Markov chain Monte Carlo (MCMC) algorithms offer various strategies for sampling; the Hamiltonian Monte Carlo (HMC) family of samplers are MCMC algorithms which often exhibit improved mixing properties. The recently introduced magnetic HMC, a generalization of HMC motivated by the physics of particles influenced by magnetic field forces, has been demonstrated to improve the performance of HMC. In many applications, one wishes to sample from a distribution restricted to a constrained set, often manifested as an embedded manifold (for example, the surface of a sphere). We introduce magnetic manifold HMC, an HMC algorithm on embedded manifolds motivated by the physics of particles constrained to a manifold and moving under magnetic field forces. We discuss the theoretical properties of magnetic Hamiltonian dynamics on manifolds, and introduce a reversible and symplectic integrator for the HMC updates. We demonstrate that magnetic manifold HMC produces favorable sampling behaviors relative to the canonical variant of manifold-constrained HMC.},
note = {arXiv: 2010.07753},
keywords = {Algorithms, Computer Science - Machine Learning, HMC, Manifolds, MCMC, Statistics - Machine Learning},
pubstate = {published},
tppubtype = {techreport}
}

Markov chain Monte Carlo (MCMC) algorithms offer various strategies for sampling; the Hamiltonian Monte Carlo (HMC) family of samplers are MCMC algorithms which often exhibit improved mixing properties. The recently introduced magnetic HMC, a generalization of HMC motivated by the physics of particles influenced by magnetic field forces, has been demonstrated to improve the performance of HMC. In many applications, one wishes to sample from a distribution restricted to a constrained set, often manifested as an embedded manifold (for example, the surface of a sphere). We introduce magnetic manifold HMC, an HMC algorithm on embedded manifolds motivated by the physics of particles constrained to a manifold and moving under magnetic field forces. We discuss the theoretical properties of magnetic Hamiltonian dynamics on manifolds, and introduce a reversible and symplectic integrator for the HMC updates. We demonstrate that magnetic manifold HMC produces favorable sampling behaviors relative to the canonical variant of manifold-constrained HMC.

@techreport{katz_spectral_2020,
title = {Spectral Flow on the Manifold of SPD Matrices for Multimodal Data Processing},
author = {Ori Katz and Roy R Lederman and Ronen Talmon},
url = {http://arxiv.org/abs/2009.08062},
year = {2020},
date = {2020-09-01},
urldate = {2020-11-25},
abstract = {In this paper, we consider data acquired by multimodal sensors capturing complementary aspects and features of a measured phenomenon. We focus on a scenario in which the measurements share mutual sources of variability but might also be contaminated by other measurement-specific sources such as interferences or noise. Our approach combines manifold learning, which is a class of nonlinear data-driven dimension reduction methods, with the well-known Riemannian geometry of symmetric and positive-definite (SPD) matrices. Manifold learning typically includes the spectral analysis of a kernel built from the measurements. Here, we take a different approach, utilizing the Riemannian geometry of the kernels. In particular, we study the way the spectrum of the kernels changes along geodesic paths on the manifold of SPD matrices. We show that this change enables us, in a purely unsupervised manner, to derive a compact, yet informative, description of the relations between the measurements, in terms of their underlying components. Based on this result, we present new algorithms for extracting the common latent components and for identifying common and measurement-specific components.},
note = {arXiv: 2009.08062},
keywords = {Common variable, Computer Science - Machine Learning, Manifold Learning, Multi-view, multimodal, SPD Matrices, Statistics - Machine Learning},
pubstate = {published},
tppubtype = {techreport}
}

In this paper, we consider data acquired by multimodal sensors capturing complementary aspects and features of a measured phenomenon. We focus on a scenario in which the measurements share mutual sources of variability but might also be contaminated by other measurement-specific sources such as interferences or noise. Our approach combines manifold learning, which is a class of nonlinear data-driven dimension reduction methods, with the well-known Riemannian geometry of symmetric and positive-definite (SPD) matrices. Manifold learning typically includes the spectral analysis of a kernel built from the measurements. Here, we take a different approach, utilizing the Riemannian geometry of the kernels. In particular, we study the way the spectrum of the kernels changes along geodesic paths on the manifold of SPD matrices. We show that this change enables us, in a purely unsupervised manner, to derive a compact, yet informative, description of the relations between the measurements, in terms of their underlying components. Based on this result, we present new algorithms for extracting the common latent components and for identifying common and measurement-specific components.

@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}
}

@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.

@article{bandeira_non-unique_2020,
title = {Non-unique games over compact groups and orientation estimation in cryo-EM},
author = {Afonso S Bandeira and Yutong Chen and Roy R Lederman and Amit Singer},
url = {https://iopscience.iop.org/article/10.1088/1361-6420/ab7d2c},
doi = {10.1088/1361-6420/ab7d2c},
issn = {0266-5611, 1361-6420},
year = {2020},
date = {2020-01-01},
urldate = {2020-08-13},
journal = {Inverse Problems},
volume = {36},
number = {6},
pages = {064002},
keywords = {Algorithms, cryo-EM, Non-unique games, Representation Theory},
pubstate = {published},
tppubtype = {article}
}

@techreport{brofos_non-canonical_2020,
title = {Non-Canonical Hamiltonian Monte Carlo},
author = {James A Brofos and Roy R Lederman},
url = {http://arxiv.org/abs/2008.08191},
year = {2020},
date = {2020-01-01},
urldate = {2020-11-25},
abstract = {Hamiltonian Monte Carlo is typically based on the assumption of an underlying canonical symplectic structure. Numerical integrators designed for the canonical structure are incompatible with motion generated by non-canonical dynamics. These non-canonical dynamics, motivated by examples in physics and symplectic geometry, correspond to techniques such as preconditioning which are routinely used to improve algorithmic performance. Indeed, recently, a special case of non-canonical structure, magnetic Hamiltonian Monte Carlo, was demonstrated to provide advantageous sampling properties. We present a framework for Hamiltonian Monte Carlo using non-canonical symplectic structures. Our experimental results demonstrate sampling advantages associated to Hamiltonian Monte Carlo with non-canonical structure. To summarize our contributions: (i) we develop non-canonical HMC from foundations in symplectic geomtry; (ii) we construct an HMC procedure using implicit integration that satisfies the detailed balance; (iii) we propose to accelerate the sampling using an textbackslashem approximate explicit methodology; (iv) we study two novel, randomly-generated non-canonical structures: magnetic momentum and the coupled magnet structure, with implicit and explicit integration.},
note = {arXiv: 2008.08191},
keywords = {Algorithms, Computer Science - Machine Learning, HMC, MCMC, Statistics - Machine Learning},
pubstate = {published},
tppubtype = {techreport}
}

Hamiltonian Monte Carlo is typically based on the assumption of an underlying canonical symplectic structure. Numerical integrators designed for the canonical structure are incompatible with motion generated by non-canonical dynamics. These non-canonical dynamics, motivated by examples in physics and symplectic geometry, correspond to techniques such as preconditioning which are routinely used to improve algorithmic performance. Indeed, recently, a special case of non-canonical structure, magnetic Hamiltonian Monte Carlo, was demonstrated to provide advantageous sampling properties. We present a framework for Hamiltonian Monte Carlo using non-canonical symplectic structures. Our experimental results demonstrate sampling advantages associated to Hamiltonian Monte Carlo with non-canonical structure. To summarize our contributions: (i) we develop non-canonical HMC from foundations in symplectic geomtry; (ii) we construct an HMC procedure using implicit integration that satisfies the detailed balance; (iii) we propose to accelerate the sampling using an textbackslashem approximate explicit methodology; (iv) we study two novel, randomly-generated non-canonical structures: magnetic momentum and the coupled magnet structure, with implicit and explicit integration.

@inproceedings{brofos_bias-variance_2019,
title = {A Bias-Variance Decomposition for Bayesian Deep Learning},
author = {James A Brofos and Rui Shu and Roy R Lederman},
year = {2019},
date = {2019-12-01},
pages = {14},
abstract = {We exhibit a decomposition of the Kullback-Leibler divergence into terms corresponding to bias, variance, and irreducible error. Our particular focus in this work is Bayesian deep learning and in this domain we illustrate the application of this decomposition to adversarial example identiﬁcation, to image segmentation, and to malware detection. We empirically demonstrate qualitative similarities between the variance decomposition and mutual information.},
keywords = {Bayesian Deep Learning, Bayesian Inference, Deep Learning},
pubstate = {published},
tppubtype = {inproceedings}
}

We exhibit a decomposition of the Kullback-Leibler divergence into terms corresponding to bias, variance, and irreducible error. Our particular focus in this work is Bayesian deep learning and in this domain we illustrate the application of this decomposition to adversarial example identiﬁcation, to image segmentation, and to malware detection. We empirically demonstrate qualitative similarities between the variance decomposition and mutual information.

@techreport{lederman_extreme_2019,
title = {Extreme Values of the Fiedler Vector on Trees},
author = {Roy R Lederman and S Steinerberger},
url = {http://arxiv.org/abs/1912.08327},
year = {2019},
date = {2019-12-01},
urldate = {2020-08-13},
abstract = {Let $G$ be a connected tree on $n$ vertices and let $L = D-A$ denote the Laplacian matrix on $G$. The second-smallest eigenvalue $textbackslashlambda_2(G) textgreater 0$, also known as the algebraic connectivity, as well as the associated eigenvector $textbackslashphi_2$ have been of substantial interest. We investigate the question of when the maxima and minima of $textbackslashphi_2$ are assumed at the endpoints of the longest path in $G$. Our results also apply to more general graphs that `behave globally' like a tree but can exhibit more complicated local structure. The crucial new ingredient is a reproducing formula for the eigenvector $textbackslashphi_k$.},
note = {arXiv: 1912.08327},
keywords = {Computer Science - Discrete Mathematics, Graph Theory, Mathematics - Combinatorics, Mathematics - Spectral Theory},
pubstate = {published},
tppubtype = {techreport}
}

Let $G$ be a connected tree on $n$ vertices and let $L = D-A$ denote the Laplacian matrix on $G$. The second-smallest eigenvalue $textbackslashlambda_2(G) textgreater 0$, also known as the algebraic connectivity, as well as the associated eigenvector $textbackslashphi_2$ have been of substantial interest. We investigate the question of when the maxima and minima of $textbackslashphi_2$ are assumed at the endpoints of the longest path in $G$. Our results also apply to more general graphs that `behave globally' like a tree but can exhibit more complicated local structure. The crucial new ingredient is a reproducing formula for the eigenvector $textbackslashphi_k$.

@article{lederman_learning_2018,
title = {Learning the geometry of common latent variables using alternating-diffusion},
author = {Roy R Lederman and Ronen Talmon},
url = {http://www.sciencedirect.com/science/article/pii/S1063520315001190},
doi = {10.1016/j.acha.2015.09.002},
issn = {1063-5203},
year = {2018},
date = {2018-01-01},
urldate = {2020-08-13},
journal = {Applied and Computational Harmonic Analysis},
volume = {44},
number = {3},
pages = {509--536},
abstract = {One of the challenges in data analysis is to distinguish between different sources of variability manifested in data. In this paper, we consider the case of multiple sensors measuring the same physical phenomenon, such that the properties of the physical phenomenon are manifested as a hidden common source of variability (which we would like to extract), while each sensor has its own sensor-specific effects (hidden variables which we would like to suppress); the relations between the measurements and the hidden variables are unknown. We present a data-driven method based on alternating products of diffusion operators and show that it extracts the common source of variability. Moreover, we show that it extracts the common source of variability in a multi-sensor experiment as if it were a standard manifold learning algorithm used to analyze a simple single-sensor experiment, in which the common source of variability is the only source of variability.},
keywords = {Algorithms, Alternating Diffusion, Alternating-diffusion, Common variable, diffusion maps, Diffusion-maps, Multi-view, multimodal, Multimodal analysis},
pubstate = {published},
tppubtype = {article}
}

One of the challenges in data analysis is to distinguish between different sources of variability manifested in data. In this paper, we consider the case of multiple sensors measuring the same physical phenomenon, such that the properties of the physical phenomenon are manifested as a hidden common source of variability (which we would like to extract), while each sensor has its own sensor-specific effects (hidden variables which we would like to suppress); the relations between the measurements and the hidden variables are unknown. We present a data-driven method based on alternating products of diffusion operators and show that it extracts the common source of variability. Moreover, we show that it extracts the common source of variability in a multi-sensor experiment as if it were a standard manifold learning algorithm used to analyze a simple single-sensor experiment, in which the common source of variability is the only source of variability.

@techreport{aldroubi_dynamical_2018,
title = {Dynamical sampling with additive random noise},
author = {Akram Aldroubi and Longxiu Huang and Ilya Krishtal and Akos Ledeczi and Roy R Lederman and Peter Volgyesi},
url = {http://arxiv.org/abs/1807.10866},
year = {2018},
date = {2018-01-01},
urldate = {2020-08-13},
number = {arXiv:1807.10866 [math]},
abstract = {Dynamical sampling deals with signals that evolve in time under the action of a linear operator. The purpose of the present paper is to analyze the performance of the basic dynamical sampling algorithms in the finite dimensional case and study the impact of additive noise. The algorithms are implemented and tested on synthetic and real data sets, and denoising techniques are integrated to mitigate the effect of the noise. We also develop theoretical and numerical results that validate the algorithm for recovering the driving operators, which are defined via a real symmetric convolution.},
note = {arXiv: 1807.10866},
keywords = {Mathematics - Numerical Analysis},
pubstate = {published},
tppubtype = {techreport}
}

Dynamical sampling deals with signals that evolve in time under the action of a linear operator. The purpose of the present paper is to analyze the performance of the basic dynamical sampling algorithms in the finite dimensional case and study the impact of additive noise. The algorithms are implemented and tested on synthetic and real data sets, and denoising techniques are integrated to mitigate the effect of the noise. We also develop theoretical and numerical results that validate the algorithm for recovering the driving operators, which are defined via a real symmetric convolution.

@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.

@techreport{lederman_numerical_2017,
title = {Numerical Algorithms for the Computation of Generalized Prolate Spheroidal Functions},
author = {Roy R Lederman},
url = {https://arxiv.org/abs/1710.02874v1},
year = {2017},
date = {2017-10-01},
urldate = {2020-08-13},
abstract = {Generalized Prolate Spheroidal Functions (GPSF) are the eigenfunctions of the
truncated Fourier transform, restricted to D-dimensional balls in the spatial
domain and frequency domain. Despite their useful properties in many
applications, GPSFs are often replaced by crude approximations. The purpose of
this paper is to review the elements of computing GPSFs and associated
eigenvalues. This paper is accompanied by open-source code.},
keywords = {Algorithms, cryo-EM, Fourier Transform, Numerical Analysis, Prolate, Slepian, Software},
pubstate = {published},
tppubtype = {techreport}
}

Generalized Prolate Spheroidal Functions (GPSF) are the eigenfunctions of the
truncated Fourier transform, restricted to D-dimensional balls in the spatial
domain and frequency domain. Despite their useful properties in many
applications, GPSFs are often replaced by crude approximations. The purpose of
this paper is to review the elements of computing GPSFs and associated
eigenvalues. This paper is accompanied by open-source code.

@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.

@article{stanton_ritornello_2017,
title = {Ritornello: high fidelity control-free chromatin immunoprecipitation peak calling},
author = {Kelly P Stanton and Jiaqi Jin and Roy R Lederman and Sherman M Weissman and Yuval Kluger},
url = {https://academic.oup.com/nar/article/45/21/e173/4157402},
doi = {10.1093/nar/gkx799},
issn = {0305-1048},
year = {2017},
date = {2017-01-01},
urldate = {2020-08-13},
journal = {Nucleic Acids Research},
volume = {45},
number = {21},
pages = {e173--e173},
abstract = {Abstract. With the advent of next generation high-throughput DNA sequencing technologies, omics experiments have become the mainstay for studying diverse biolo},
note = {Publisher: Oxford Academic},
keywords = {DNA sequencing, Sequencing, Software},
pubstate = {published},
tppubtype = {article}
}

Abstract. With the advent of next generation high-throughput DNA sequencing technologies, omics experiments have become the mainstay for studying diverse biolo

@inproceedings{aldroubi_dynamical_2017,
title = {Dynamical sampling with random noise},
author = {Akram Aldroubi and L Huang and I Krishtal and Roy R Lederman},
doi = {10.1109/SAMPTA.2017.8024372},
year = {2017},
date = {2017-01-01},
booktitle = {2017 International Conference on Sampling Theory and Applications (SampTA)},
pages = {409--412},
abstract = {In this paper we consider a system of dynamical sampling, i.e. sampling a signal f that evolves in time under the action of an evolution operator A. We discuss the error in the recovery of the original signal when the samples are corrupted by additive, independent and identically distributed (i.i.d) noise. We focus on the study of the mean squared error E(∥ϵn∥22) between the original signal and the reconstructed signal obtained by solving a least squares problem. In the theoretical part, we give a formula for E(∥ϵn∥22) and prove that E(∥ϵn∥22) decreases as the number of the samples increases. In addition, we discuss several numerical experiments that verify the theoretical results.},
keywords = {Dynamical Sampling, evolution operator, signal reconstruction, signal recovery, signal sampling},
pubstate = {published},
tppubtype = {inproceedings}
}

In this paper we consider a system of dynamical sampling, i.e. sampling a signal f that evolves in time under the action of an evolution operator A. We discuss the error in the recovery of the original signal when the samples are corrupted by additive, independent and identically distributed (i.i.d) noise. We focus on the study of the mean squared error E(∥ϵn∥22) between the original signal and the reconstructed signal obtained by solving a least squares problem. In the theoretical part, we give a formula for E(∥ϵn∥22) and prove that E(∥ϵn∥22) decreases as the number of the samples increases. In addition, we discuss several numerical experiments that verify the theoretical results.

@techreport{lederman_representation_2016,
title = {A Representation Theory Perspective on Simultaneous Alignment and Classification},
author = {Roy R Lederman and Amit Singer},
url = {http://arxiv.org/abs/1607.03464},
year = {2016},
date = {2016-07-01},
urldate = {2021-01-22},
number = {arXiv:1607.03464 [cs, math]},
abstract = {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.},
note = {arXiv: 1607.03464},
keywords = {Algorithms, Computer Science - Computer Vision and Pattern Recognition, cryo-EM, Mathematics - Optimization and Control, Representation Theory},
pubstate = {published},
tppubtype = {techreport}
}

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.

@techreport{lederman_stability_2016,
title = {Stability Estimates for Truncated Fourier and Laplace Transforms},
author = {Roy R Lederman and Stefan Steinerberger},
url = {https://arxiv.org/abs/1605.03866v1},
year = {2016},
date = {2016-05-01},
urldate = {2020-08-13},
number = {arXiv:1605.03866},
abstract = {We prove sharp stability estimates for the Truncated Laplace Transform and
Truncated Fourier Transform. The argument combines an approach recently
introduced by Alaifari, Pierce and the second author for the truncated Hilbert
transform with classical results of Bertero, Grünbaum, Landau, Pollak and
Slepian. In particular, we prove there is a universal constant $c textgreater0$ such that
for all $f textbackslashin Ltextasciicircum2(textbackslashmathbbR)$ with compact support in $[-1,1]$ normalized to $textbackslashtextbarftextbackslashtextbar_Ltextasciicircum2[-1,1] = 1$ $$ textbackslashint_-1textasciicircum1textbartextbackslashwidehatf(ξ)textbartextasciicircum2dξ textbackslashgtrsim
textbackslashleft(ctextbackslashlefttextbackslashtextbarf_x textbackslashrighttextbackslashtextbar_Ltextasciicircum2[-1,1] textbackslashright)textasciicircumvphantom- ctextbackslashlefttextbackslashtextbarf_x
textbackslashrighttextbackslashtextbar_Ltextasciicircum2[-1,1]vphantom$$ The inequality is sharp in the sense that there is an
infinite sequence of orthonormal counterexamples if $c$ is chosen too small.
The question whether and to which extent similar inequalities hold for generic
families of integral operators remains open.},
keywords = {Laplace Transform},
pubstate = {published},
tppubtype = {techreport}
}

We prove sharp stability estimates for the Truncated Laplace Transform and
Truncated Fourier Transform. The argument combines an approach recently
introduced by Alaifari, Pierce and the second author for the truncated Hilbert
transform with classical results of Bertero, Grünbaum, Landau, Pollak and
Slepian. In particular, we prove there is a universal constant $c textgreater0$ such that
for all $f textbackslashin Ltextasciicircum2(textbackslashmathbbR)$ with compact support in $[-1,1]$ normalized to $textbackslashtextbarftextbackslashtextbar_Ltextasciicircum2[-1,1] = 1$ $$ textbackslashint_-1textasciicircum1textbartextbackslashwidehatf(ξ)textbartextasciicircum2dξ textbackslashgtrsim
textbackslashleft(ctextbackslashlefttextbackslashtextbarf_x textbackslashrighttextbackslashtextbar_Ltextasciicircum2[-1,1] textbackslashright)textasciicircumvphantom- ctextbackslashlefttextbackslashtextbarf_x
textbackslashrighttextbackslashtextbar_Ltextasciicircum2[-1,1]vphantom$$ The inequality is sharp in the sense that there is an
infinite sequence of orthonormal counterexamples if $c$ is chosen too small.
The question whether and to which extent similar inequalities hold for generic
families of integral operators remains open.

@techreport{shaham_common_2015,
title = {Common Variable Learning and Invariant Representation Learning using Siamese Neural Networks},
author = {Uri Shaham and Roy R Lederman},
url = {https://arxiv.org/abs/1512.08806v3},
year = {2015},
date = {2015-12-01},
urldate = {2020-08-13},
abstract = {We consider the statistical problem of learning common source of variability
in data which are synchronously captured by multiple sensors, and demonstrate
that Siamese neural networks can be naturally applied to this problem. This
approach is useful in particular in exploratory, data-driven applications,
where neither a model nor label information is available. In recent years, many
researchers have successfully applied Siamese neural networks to obtain an
embedding of data which corresponds to a "semantic similarity". We present an
interpretation of this "semantic similarity" as learning of equivalence
classes. We discuss properties of the embedding obtained by Siamese networks
and provide empirical results that demonstrate the ability of Siamese networks
to learn common variability.},
keywords = {Common variable, Deep Learning, Multi-view},
pubstate = {published},
tppubtype = {techreport}
}

We consider the statistical problem of learning common source of variability
in data which are synchronously captured by multiple sensors, and demonstrate
that Siamese neural networks can be naturally applied to this problem. This
approach is useful in particular in exploratory, data-driven applications,
where neither a model nor label information is available. In recent years, many
researchers have successfully applied Siamese neural networks to obtain an
embedding of data which corresponds to a "semantic similarity". We present an
interpretation of this "semantic similarity" as learning of equivalence
classes. We discuss properties of the embedding obtained by Siamese networks
and provide empirical results that demonstrate the ability of Siamese networks
to learn common variability.

@inproceedings{lederman_alternating_2015,
title = {Alternating diffusion for common manifold learning with application to sleep stage assessment},
author = {Roy R Lederman and Ronen Talmon and Hau-tieng Wu and Yu-Lun Lo and Ronald R Coifman},
doi = {10.1109/ICASSP.2015.7179075},
year = {2015},
date = {2015-01-01},
booktitle = {2015 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)},
pages = {5758--5762},
abstract = {In this paper, we address the problem of multimodal signal processing and present a manifold learning method to extract the common source of variability from multiple measurements. This method is based on alternating-diffusion and is particularly adapted to time series. We show that the common source of variability is extracted from multiple sensors as if it were the only source of variability, extracted by a standard manifold learning method from a single sensor, without the influence of the sensor-specific variables. In addition, we present application to sleep stage assessment. We demonstrate that, indeed, through alternating-diffusion, the sleep information hidden inside multimodal respiratory signals can be better captured compared to single-modal methods.},
note = {ISSN: 2379-190X},
keywords = {Alternating Diffusion, Common variable, diffusion maps, Kernel, learning (artificial intelligence), Manifolds, multimodal, multimodal respiratory signals, multimodal signal processing, Physiology, Sensitivity, Sensor phenomena and characterization, signal processing, sleep, sleep stage assessment, standard manifold learning method, time series},
pubstate = {published},
tppubtype = {inproceedings}
}

In this paper, we address the problem of multimodal signal processing and present a manifold learning method to extract the common source of variability from multiple measurements. This method is based on alternating-diffusion and is particularly adapted to time series. We show that the common source of variability is extracted from multiple sensors as if it were the only source of variability, extracted by a standard manifold learning method from a single sensor, without the influence of the sensor-specific variables. In addition, we present application to sleep stage assessment. We demonstrate that, indeed, through alternating-diffusion, the sleep information hidden inside multimodal respiratory signals can be better captured compared to single-modal methods.

@article{lederman_random-permutations-based_2013,
title = {A random-permutations-based approach to fast read alignment},
author = {Roy R Lederman},
url = {https://doi.org/10.1186/1471-2105-14-S5-S8},
doi = {10.1186/1471-2105-14-S5-S8},
issn = {1471-2105},
year = {2013},
date = {2013-04-01},
urldate = {2021-10-26},
journal = {BMC Bioinformatics},
volume = {14},
number = {5},
pages = {S8},
abstract = {Read alignment is a computational bottleneck in some sequencing projects. Most of the existing software packages for read alignment are based on two algorithmic approaches: prefix-trees and hash-tables. We propose a new approach to read alignment using random permutations of strings.},
keywords = {Algorithm, DNA sequencing, Fast algorithms, Neighbor Search, Random Permutation, Reference Genome, Reference Library, Search Problem},
pubstate = {published},
tppubtype = {article}
}

Read alignment is a computational bottleneck in some sequencing projects. Most of the existing software packages for read alignment are based on two algorithmic approaches: prefix-trees and hash-tables. We propose a new approach to read alignment using random permutations of strings.

@techreport{lederman_permutations-based_2013,
title = {A permutations-based algorithm for fast alignment of long paired-end reads},
author = {Roy R Lederman},
year = {2013},
date = {2013-04-01},
number = {YALEU/DCS/TR-1474},
pages = {11},
institution = {Yale CS},
keywords = {Algorithms, DNA sequencing, Fast algorithms, Randomized algorithms, Sequencing},
pubstate = {published},
tppubtype = {techreport}
}