Herreros, D; Lederman, Roy R; Krieger, J; JiménezMoreno, A; Martínez, M; Myška, D; Strelak, D; Filipovic, J; Bahar, I; Carazo, J M; Sanchez, C O S Approximating deformation fields for the analysis of continuous heterogeneity of biological macromolecules by 3D Zernike polynomials Journal Article IUCrJ, 8 (6), 2021, ISSN: 20522525, (Number: 6
Publisher: International Union of Crystallography). Abstract  Links  BibTeX  Tags: cryoEM, heterogeneity, Zernike @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 JiménezMoreno 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 = {20522525},
year = {2021},
date = {20211101},
urldate = {20211026},
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 = {cryoEM, 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 Continuous heterogeneity analysis of CryoEM images through Zernike polynomials and spherical harmonics Journal Article Microscopy and Microanalysis, 27 (S1), pp. 1680–1682, 2021, ISSN: 14319276, 14358115, (Publisher: Cambridge University Press). Abstract  Links  BibTeX  Tags: cryoEM, heterogeneity, Zernike @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/microscopyandmicroanalysis/article/continuousheterogeneityanalysisofcryoemimagesthroughzernikepolynomialsandsphericalharmonics/2A8C58651F413C8A0D66071CB4BC9AAD},
doi = {10.1017/S1431927621006176},
issn = {14319276, 14358115},
year = {2021},
date = {20210801},
urldate = {20210803},
journal = {Microscopy and Microanalysis},
volume = {27},
number = {S1},
pages = {16801682},
abstract = {//static.cambridge.org/content/id/urn%3Acambridge.org%3Aid%3Aarticle%3AS1431927621006176/resource/name/firstPageS1431927621006176a.jpg},
note = {Publisher: Cambridge University Press},
keywords = {cryoEM, heterogeneity, Zernike},
pubstate = {published},
tppubtype = {article}
}
//static.cambridge.org/content/id/urn%3Acambridge.org%3Aid%3Aarticle%3AS1431927621006176/resource/name/firstPageS1431927621006176a.jpg 
Brofos, James; Lederman, Roy R Evaluating the Implicit Midpoint Integrator for Riemannian Hamiltonian Monte Carlo Inproceedings International Conference on Machine Learning, pp. 1072–1081, PMLR, 2021, (ISSN: 26403498). Links  BibTeX  Tags: HMC, Manifolds, MCMC, Numerical Analysis @inproceedings{brofos_evaluating_2021,
title = {Evaluating the Implicit Midpoint Integrator for Riemannian Hamiltonian Monte Carlo},
author = {James Brofos and Roy R Lederman},
url = {http://proceedings.mlr.press/v139/brofos21a.html},
year = {2021},
date = {20210701},
urldate = {20210714},
booktitle = {International Conference on Machine Learning},
pages = {10721081},
publisher = {PMLR},
note = {ISSN: 26403498},
keywords = {HMC, Manifolds, MCMC, Numerical Analysis},
pubstate = {published},
tppubtype = {inproceedings}
}

Brofos, James A; Brubaker, Marcus A; Lederman, Roy R Manifold Density Estimation via Generalized Dequantization Technical Report 2021, (arXiv: 2102.07143). Abstract  Links  BibTeX  Tags: Algorithms, Computer Science  Machine Learning, Density estimation, Manifolds, Statistics  Machine Learning @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 = {20210701},
urldate = {20210714},
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 wellmodeled 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 wellmodeled 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. 
Fan, Zhou; Lederman, Roy R; Sun, Yi; Wang, Tianhao; Xu, Sheng Maximum likelihood for highnoise group orbit estimation and singleparticle cryoEM Technical Report 2021, (arXiv: 2107.01305). Abstract  Links  BibTeX  Tags: Computer Science  Information Theory, Mathematics  Optimization and Control, Mathematics  Statistics Theory @techreport{fan_maximum_2021,
title = {Maximum likelihood for highnoise group orbit estimation and singleparticle cryoEM},
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 = {20210701},
urldate = {20210714},
abstract = {Motivated by applications to singleparticle cryoelectron microscopy (cryoEM), 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 loglikelihood landscape to a sequence of methodofmoments 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 cryoEM as introduced by Bandeira, BlumSmith, 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 lowdimensional approximations of the electric potential maps of two small protein molecules, we empirically verify that the noisescalings 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 singleparticle cryoelectron microscopy (cryoEM), 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 loglikelihood landscape to a sequence of methodofmoments 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 cryoEM as introduced by Bandeira, BlumSmith, 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 lowdimensional approximations of the electric potential maps of two small protein molecules, we empirically verify that the noisescalings of the Fisher information eigenvalues conform with these theoretical predictions over a range of SNR, in a model of $SO(3)$ rotations without projection. 
Brofos, James A; Lederman, Roy R Magnetic Manifold Hamiltonian Monte Carlo Technical Report 2020, (arXiv: 2010.07753). Abstract  Links  BibTeX  Tags: Algorithms, Computer Science  Machine Learning, HMC, Manifolds, MCMC, Statistics  Machine Learning @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 = {20201001},
urldate = {20201125},
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 manifoldconstrained 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 manifoldconstrained HMC. 
Katz, Ori; Lederman, Roy R; Talmon, Ronen Spectral Flow on the Manifold of SPD Matrices for Multimodal Data Processing Technical Report 2020, (arXiv: 2009.08062). Abstract  Links  BibTeX  Tags: Common variable, Computer Science  Machine Learning, Manifold Learning, Multiview, multimodal, SPD Matrices, Statistics  Machine Learning @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 = {20200901},
urldate = {20201125},
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 measurementspecific sources such as interferences or noise. Our approach combines manifold learning, which is a class of nonlinear datadriven dimension reduction methods, with the wellknown Riemannian geometry of symmetric and positivedefinite (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 measurementspecific components.},
note = {arXiv: 2009.08062},
keywords = {Common variable, Computer Science  Machine Learning, Manifold Learning, Multiview, 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 measurementspecific sources such as interferences or noise. Our approach combines manifold learning, which is a class of nonlinear datadriven dimension reduction methods, with the wellknown Riemannian geometry of symmetric and positivedefinite (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 measurementspecific components. 
Lederman, Roy R; Andén, Joakim; Singer, Amit Hypermolecules: on the representation and recovery of dynamical structures for applications in flexible macromolecules in cryoEM Journal Article Inverse Problems, 36 (4), pp. 044005, 2020, ISSN: 02665611, 13616420. Links  BibTeX  Tags: cryoEM, heterogeneity, HyperMolecules, MCMC, Variational inference @article{lederman_hypermolecules_2020,
title = {Hypermolecules: on the representation and recovery of dynamical structures for applications in flexible macromolecules in cryoEM},
author = {Roy R Lederman and Joakim Andén and Amit Singer},
url = {https://iopscience.iop.org/article/10.1088/13616420/ab5ede},
doi = {10.1088/13616420/ab5ede},
issn = {02665611, 13616420},
year = {2020},
date = {20200401},
urldate = {20200813},
journal = {Inverse Problems},
volume = {36},
number = {4},
pages = {044005},
keywords = {cryoEM, 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: 10635203. Abstract  Links  BibTeX  Tags: Algorithms, Alignment, Classification, cryoEM, Graphcut, 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 = {10635203},
year = {2020},
date = {20200101},
urldate = {20210122},
journal = {Applied and Computational Harmonic Analysis},
volume = {49},
number = {3},
pages = {10011024},
abstract = {Single particle cryoelectron microscopy (EM) is a method for determining the 3D structure of macromolecules from many noisy 2D 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 nonunique 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 cryoEM 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 nonunique games to the problem of simultaneous alignment and classification with the goal of simultaneously classifying cryoEM images and aligning them within their classes.},
keywords = {Algorithms, Alignment, Classification, cryoEM, Graphcut, heterogeneity, Heterogeneous multireference alignment, Representation Theory, Rotation group, SDP, Synchronization},
pubstate = {published},
tppubtype = {article}
}
Single particle cryoelectron microscopy (EM) is a method for determining the 3D structure of macromolecules from many noisy 2D 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 nonunique 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 cryoEM 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 nonunique games to the problem of simultaneous alignment and classification with the goal of simultaneously classifying cryoEM images and aligning them within their classes. 
Bandeira, Afonso S; Chen, Yutong; Lederman, Roy R; Singer, Amit Nonunique games over compact groups and orientation estimation in cryoEM Journal Article Inverse Problems, 36 (6), pp. 064002, 2020, ISSN: 02665611, 13616420. Links  BibTeX  Tags: Algorithms, cryoEM, Nonunique games, Representation Theory @article{bandeira_nonunique_2020,
title = {Nonunique games over compact groups and orientation estimation in cryoEM},
author = {Afonso S Bandeira and Yutong Chen and Roy R Lederman and Amit Singer},
url = {https://iopscience.iop.org/article/10.1088/13616420/ab7d2c},
doi = {10.1088/13616420/ab7d2c},
issn = {02665611, 13616420},
year = {2020},
date = {20200101},
urldate = {20200813},
journal = {Inverse Problems},
volume = {36},
number = {6},
pages = {064002},
keywords = {Algorithms, cryoEM, Nonunique games, Representation Theory},
pubstate = {published},
tppubtype = {article}
}

Brofos, James A; Lederman, Roy R NonCanonical Hamiltonian Monte Carlo Technical Report 2020, (arXiv: 2008.08191). Abstract  Links  BibTeX  Tags: Algorithms, Computer Science  Machine Learning, HMC, MCMC, Statistics  Machine Learning @techreport{brofos_noncanonical_2020,
title = {NonCanonical Hamiltonian Monte Carlo},
author = {James A Brofos and Roy R Lederman},
url = {http://arxiv.org/abs/2008.08191},
year = {2020},
date = {20200101},
urldate = {20201125},
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 noncanonical dynamics. These noncanonical 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 noncanonical structure, magnetic Hamiltonian Monte Carlo, was demonstrated to provide advantageous sampling properties. We present a framework for Hamiltonian Monte Carlo using noncanonical symplectic structures. Our experimental results demonstrate sampling advantages associated to Hamiltonian Monte Carlo with noncanonical structure. To summarize our contributions: (i) we develop noncanonical 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, randomlygenerated noncanonical 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 noncanonical dynamics. These noncanonical 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 noncanonical structure, magnetic Hamiltonian Monte Carlo, was demonstrated to provide advantageous sampling properties. We present a framework for Hamiltonian Monte Carlo using noncanonical symplectic structures. Our experimental results demonstrate sampling advantages associated to Hamiltonian Monte Carlo with noncanonical structure. To summarize our contributions: (i) we develop noncanonical 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, randomlygenerated noncanonical structures: magnetic momentum and the coupled magnet structure, with implicit and explicit integration. 
Brofos, James A; Shu, Rui; Lederman, Roy R A BiasVariance Decomposition for Bayesian Deep Learning Inproceedings pp. 14, 2019. Abstract  BibTeX  Tags: Bayesian Deep Learning, Bayesian Inference, Deep Learning @inproceedings{brofos_biasvariance_2019,
title = {A BiasVariance Decomposition for Bayesian Deep Learning},
author = {James A Brofos and Rui Shu and Roy R Lederman},
year = {2019},
date = {20191201},
pages = {14},
abstract = {We exhibit a decomposition of the KullbackLeibler 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 KullbackLeibler 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. 
Lederman, Roy R; Steinerberger, S Extreme Values of the Fiedler Vector on Trees Technical Report 2019, (arXiv: 1912.08327). Abstract  Links  BibTeX  Tags: Computer Science  Discrete Mathematics, Graph Theory, Mathematics  Combinatorics, Mathematics  Spectral Theory @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 = {20191201},
urldate = {20200813},
abstract = {Let $G$ be a connected tree on $n$ vertices and let $L = DA$ denote the Laplacian matrix on $G$. The secondsmallest 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 $L = DA$ denote the Laplacian matrix on $G$. The secondsmallest 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 $textbackslashphi_k$. 
Shnitzer, Tal; Lederman, Roy R; Liu, GiRen; Talmon, Ronen; Wu, HauTieng Diffusion operators for multimodal data analysis Incollection Handbook of Numerical Analysis, 20 , pp. 1–39, Elsevier, 2019, ISBN: 9780444641403. Links  BibTeX  Tags: Alternating Diffusion, BookChapter, Common variable, diffusion maps, Manifold Learning, Multiview, multimodal, Multimodal data, Sensor fusion, Shape differences @incollection{shnitzer_diffusion_2019,
title = {Diffusion operators for multimodal data analysis},
author = {Tal Shnitzer and Roy R Lederman and GiRen Liu and Ronen Talmon and HauTieng Wu},
url = {https://linkinghub.elsevier.com/retrieve/pii/S1570865919300213},
doi = {10.1016/bs.hna.2019.07.008},
isbn = {9780444641403},
year = {2019},
date = {20190101},
urldate = {20200813},
booktitle = {Handbook of Numerical Analysis},
volume = {20},
pages = {139},
publisher = {Elsevier},
keywords = {Alternating Diffusion, BookChapter, Common variable, diffusion maps, Manifold Learning, Multiview, multimodal, Multimodal data, Sensor fusion, Shape differences},
pubstate = {published},
tppubtype = {incollection}
}

Lederman, Roy R; Talmon, Ronen Learning the geometry of common latent variables using alternatingdiffusion Journal Article Applied and Computational Harmonic Analysis, 44 (3), pp. 509–536, 2018, ISSN: 10635203. Abstract  Links  BibTeX  Tags: Algorithms, Alternating Diffusion, Alternatingdiffusion, Common variable, diffusion maps, Diffusionmaps, Multiview, multimodal, Multimodal analysis @article{lederman_learning_2018,
title = {Learning the geometry of common latent variables using alternatingdiffusion},
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 = {10635203},
year = {2018},
date = {20180101},
urldate = {20200813},
journal = {Applied and Computational Harmonic Analysis},
volume = {44},
number = {3},
pages = {509536},
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 sensorspecific effects (hidden variables which we would like to suppress); the relations between the measurements and the hidden variables are unknown. We present a datadriven 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 multisensor experiment as if it were a standard manifold learning algorithm used to analyze a simple singlesensor experiment, in which the common source of variability is the only source of variability.},
keywords = {Algorithms, Alternating Diffusion, Alternatingdiffusion, Common variable, diffusion maps, Diffusionmaps, Multiview, 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 sensorspecific effects (hidden variables which we would like to suppress); the relations between the measurements and the hidden variables are unknown. We present a datadriven 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 multisensor experiment as if it were a standard manifold learning algorithm used to analyze a simple singlesensor experiment, in which the common source of variability is the only source of variability. 
Aldroubi, Akram; Huang, Longxiu; Krishtal, Ilya; Ledeczi, Akos; Lederman, Roy R; Volgyesi, Peter Dynamical sampling with additive random noise Technical Report (arXiv:1807.10866 [math]), 2018, (arXiv: 1807.10866). Abstract  Links  BibTeX  Tags: Mathematics  Numerical Analysis @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 = {20180101},
urldate = {20200813},
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. 
Shaham, Uri; Lederman, Roy R Learning by coincidence: Siamese networks and common variable learning Journal Article Pattern Recognition, 74 , pp. 52–63, 2018, ISSN: 00313203. Links  BibTeX  Tags: Common variable, Deep Learning, Multiview, multimodal, Siamese networks @article{shaham_learning_2018,
title = {Learning by coincidence: Siamese networks and common variable learning},
author = {Uri Shaham and Roy R Lederman},
url = {https://linkinghub.elsevier.com/retrieve/pii/S0031320317303588},
doi = {10.1016/j.patcog.2017.09.015},
issn = {00313203},
year = {2018},
date = {20180101},
urldate = {20200813},
journal = {Pattern Recognition},
volume = {74},
pages = {5263},
keywords = {Common variable, Deep Learning, Multiview, multimodal, Siamese networks},
pubstate = {published},
tppubtype = {article}
}

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, cryoEM, cyclic shifts, Discrete Fourier transforms, estimation theory, expectationmaximization, Gaussian mixture models, heterogeneity, heterogeneous MRA, Heterogeneous multireference alignment, Multireference alignment, Noise measurement, nonconvex 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 = {20180101},
booktitle = {2018 52nd Annual Conference on Information Sciences and Systems (CISS)},
pages = {16},
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 cryoelectron 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 loworder 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, nonconvex 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 nonconvexity, 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, cryoEM, cyclic shifts, Discrete Fourier transforms, estimation theory, expectationmaximization, Gaussian mixture models, heterogeneity, heterogeneous MRA, Heterogeneous multireference alignment, Multireference alignment, Noise measurement, nonconvex 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 cryoelectron 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 loworder 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, nonconvex 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 nonconvexity, 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 Numerical Algorithms for the Computation of Generalized Prolate Spheroidal Functions Technical Report 2017. Abstract  Links  BibTeX  Tags: Algorithms, cryoEM, Fourier Transform, Numerical Analysis, Prolate, Slepian, Software @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 = {20171001},
urldate = {20200813},
abstract = {Generalized Prolate Spheroidal Functions (GPSF) are the eigenfunctions of the
truncated Fourier transform, restricted to Ddimensional 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 opensource code.},
keywords = {Algorithms, cryoEM, 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 Ddimensional 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 opensource code. 
Lederman, Roy R; Steinerberger, Stefan Lower Bounds for Truncated Fourier and Laplace Transforms Journal Article Integral Equations and Operator Theory, 87 (4), pp. 529–543, 2017, ISSN: 0378620X, 14208989. Links  BibTeX  Tags: Fourier Transform, Laplace Transform @article{lederman_lower_2017,
title = {Lower Bounds for Truncated Fourier and Laplace Transforms},
author = {Roy R Lederman and Stefan Steinerberger},
url = {http://link.springer.com/10.1007/s000200172364z},
doi = {10.1007/s000200172364z},
issn = {0378620X, 14208989},
year = {2017},
date = {20170401},
urldate = {20200813},
journal = {Integral Equations and Operator Theory},
volume = {87},
number = {4},
pages = {529543},
keywords = {Fourier Transform, Laplace Transform},
pubstate = {published},
tppubtype = {article}
}

Lederman, Roy R; Singer, Amit Continuously heterogeneous hyperobjects in cryoEM and 3Đ 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, cryoEM, heterogeneity, HyperMolecules @techreport{lederman_continuously_2017,
title = {Continuously heterogeneous hyperobjects in cryoEM and 3Đ movies of many temporal dimensions},
author = {Roy R Lederman and Amit Singer},
url = {http://arxiv.org/abs/1704.02899},
year = {2017},
date = {20170401},
urldate = {20200813},
number = {arXiv:1704.02899 [cs]},
abstract = {Single particle cryoelectron microscopy (EM) is an increasingly popular method for determining the 3D structure of macromolecules from noisy 2D images of single macromolecules whose orientations and positions are random and unknown. One of the great opportunities in cryoEM 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 cryoEM 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, cryoEM, heterogeneity, HyperMolecules},
pubstate = {published},
tppubtype = {techreport}
}
Single particle cryoelectron microscopy (EM) is an increasingly popular method for determining the 3D structure of macromolecules from noisy 2D images of single macromolecules whose orientations and positions are random and unknown. One of the great opportunities in cryoEM 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 cryoEM 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. 
Stanton, Kelly P; Jin, Jiaqi; Lederman, Roy R; Weissman, Sherman M; Kluger, Yuval Ritornello: high fidelity controlfree chromatin immunoprecipitation peak calling Journal Article Nucleic Acids Research, 45 (21), pp. e173–e173, 2017, ISSN: 03051048, (Publisher: Oxford Academic). Abstract  Links  BibTeX  Tags: DNA sequencing, Sequencing, Software @article{stanton_ritornello_2017,
title = {Ritornello: high fidelity controlfree 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 = {03051048},
year = {2017},
date = {20170101},
urldate = {20200813},
journal = {Nucleic Acids Research},
volume = {45},
number = {21},
pages = {e173e173},
abstract = {Abstract. With the advent of next generation highthroughput 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 highthroughput DNA sequencing technologies, omics experiments have become the mainstay for studying diverse biolo 
Aldroubi, Akram; Huang, L; Krishtal, I; Lederman, Roy R Dynamical sampling with random noise Inproceedings 2017 International Conference on Sampling Theory and Applications (SampTA), pp. 409–412, 2017. Abstract  Links  BibTeX  Tags: Dynamical Sampling, evolution operator, signal reconstruction, signal recovery, signal sampling @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 = {20170101},
booktitle = {2017 International Conference on Sampling Theory and Applications (SampTA)},
pages = {409412},
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. 
Lederman, Roy R; Singer, Amit A Representation Theory Perspective on Simultaneous Alignment and Classification Technical Report (arXiv:1607.03464 [cs, math]), 2016, (arXiv: 1607.03464). Abstract  Links  BibTeX  Tags: Algorithms, Computer Science  Computer Vision and Pattern Recognition, cryoEM, Mathematics  Optimization and Control, Representation Theory @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 = {20160701},
urldate = {20210122},
number = {arXiv:1607.03464 [cs, math]},
abstract = {One of the difficulties in 3D reconstruction of molecules from images in single particle CryoElectron Microscopy (CryoEM), 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 CryoEM experiments are based on iterative approximations of the molecules in a nonconvex 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 NonUnique 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 NonUnique Games to the problem of simultaneous alignment and classification with the goal of simultaneously classifying CryoEM 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, cryoEM, Mathematics  Optimization and Control, Representation Theory},
pubstate = {published},
tppubtype = {techreport}
}
One of the difficulties in 3D reconstruction of molecules from images in single particle CryoElectron Microscopy (CryoEM), 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 CryoEM experiments are based on iterative approximations of the molecules in a nonconvex 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 NonUnique 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 NonUnique Games to the problem of simultaneous alignment and classification with the goal of simultaneously classifying CryoEM images and aligning them within their respective classes. Our proposed approach can also be extended to the case of continuous heterogeneity. 
Lederman, Roy R; Steinerberger, Stefan Stability Estimates for Truncated Fourier and Laplace Transforms Technical Report (arXiv:1605.03866), 2016. Abstract  Links  BibTeX  Tags: Laplace Transform @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 = {20160501},
urldate = {20200813},
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. 
Lederman, Roy R; Rokhlin, V On the Analytical and Numerical Properties of the Truncated Laplace Transform. Part II Journal Article SIAM Journal on Numerical Analysis, 54 (2), pp. 665–687, 2016, ISSN: 00361429, 10957170. Links  BibTeX  Tags: Algorithms, Laplace Transform, Numerical Analysis @article{lederman_analytical_2016,
title = {On the Analytical and Numerical Properties of the Truncated Laplace Transform. Part II},
author = {Roy R Lederman and V Rokhlin},
url = {http://epubs.siam.org/doi/10.1137/15M1028583},
doi = {10.1137/15M1028583},
issn = {00361429, 10957170},
year = {2016},
date = {20160101},
urldate = {20200813},
journal = {SIAM Journal on Numerical Analysis},
volume = {54},
number = {2},
pages = {665687},
keywords = {Algorithms, Laplace Transform, Numerical Analysis},
pubstate = {published},
tppubtype = {article}
}

Shaham, Uri; Lederman, Roy R Common Variable Learning and Invariant Representation Learning using Siamese Neural Networks Technical Report 2015. Abstract  Links  BibTeX  Tags: Common variable, Deep Learning, Multiview @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 = {20151201},
urldate = {20200813},
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, datadriven 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, Multiview},
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, datadriven 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. 
Lederman, Roy R; Talmon, Ronen; Wu, Hautieng; Lo, YuLun; Coifman, Ronald R Alternating diffusion for common manifold learning with application to sleep stage assessment Inproceedings 2015 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), pp. 5758–5762, 2015, (ISSN: 2379190X). Abstract  Links  BibTeX  Tags: 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 @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 Hautieng Wu and YuLun Lo and Ronald R Coifman},
doi = {10.1109/ICASSP.2015.7179075},
year = {2015},
date = {20150101},
booktitle = {2015 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)},
pages = {57585762},
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 alternatingdiffusion 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 sensorspecific variables. In addition, we present application to sleep stage assessment. We demonstrate that, indeed, through alternatingdiffusion, the sleep information hidden inside multimodal respiratory signals can be better captured compared to singlemodal methods.},
note = {ISSN: 2379190X},
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 alternatingdiffusion 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 sensorspecific variables. In addition, we present application to sleep stage assessment. We demonstrate that, indeed, through alternatingdiffusion, the sleep information hidden inside multimodal respiratory signals can be better captured compared to singlemodal methods. 
Lederman, Roy R; Rokhlin, V On the Analytical and Numerical Properties of the Truncated Laplace Transform I. Journal Article SIAM Journal on Numerical Analysis, 53 (3), pp. 1214–1235, 2015, ISSN: 00361429, 10957170. Links  BibTeX  Tags: Algorithms, Laplace Transform, Numerical Analysis @article{lederman_analytical_2015,
title = {On the Analytical and Numerical Properties of the Truncated Laplace Transform I.},
author = {Roy R Lederman and V Rokhlin},
url = {http://epubs.siam.org/doi/10.1137/140990681},
doi = {10.1137/140990681},
issn = {00361429, 10957170},
year = {2015},
date = {20150101},
urldate = {20200813},
journal = {SIAM Journal on Numerical Analysis},
volume = {53},
number = {3},
pages = {12141235},
keywords = {Algorithms, Laplace Transform, Numerical Analysis},
pubstate = {published},
tppubtype = {article}
}

Lederman, Roy R; Talmon, Ronen Common Manifold Learning Using AlternatingDiﬀusion Technical Report Yale CS (YALEU/DCS/TR1497), 2014. Links  BibTeX  Tags: AD, Algorithms, Alternating Diffusion, Manifold Learning @techreport{lederman_common_2014,
title = {Common Manifold Learning Using AlternatingDiﬀusion},
author = {Roy R Lederman and Ronen Talmon},
url = {https://cpsc.yale.edu/sites/default/files/files/tr1497.pdf},
year = {2014},
date = {20140101},
number = {YALEU/DCS/TR1497},
pages = {42},
institution = {Yale CS},
keywords = {AD, Algorithms, Alternating Diffusion, Manifold Learning},
pubstate = {published},
tppubtype = {techreport}
}

Lederman, Roy R On the Analytical and Numerical Properties of the Truncated Laplace Transform PhD Thesis Yale University, 2014, (YALEU/DCS/TR1490). BibTeX  Tags: Algorithms, Laplace Transform, Numerical Analysis @phdthesis{lederman_analytical_2014,
title = {On the Analytical and Numerical Properties of the Truncated Laplace Transform},
author = {Roy R Lederman},
year = {2014},
date = {20140101},
school = {Yale University},
note = {YALEU/DCS/TR1490},
keywords = {Algorithms, Laplace Transform, Numerical Analysis},
pubstate = {published},
tppubtype = {phdthesis}
}

Lederman, Roy R A randompermutationsbased approach to fast read alignment Journal Article BMC Bioinformatics, 14 (5), pp. S8, 2013, ISSN: 14712105. Abstract  Links  BibTeX  Tags: Algorithm, DNA sequencing, Fast algorithms, Neighbor Search, Random Permutation, Reference Genome, Reference Library, Search Problem @article{lederman_randompermutationsbased_2013,
title = {A randompermutationsbased approach to fast read alignment},
author = {Roy R Lederman},
url = {https://doi.org/10.1186/1471210514S5S8},
doi = {10.1186/1471210514S5S8},
issn = {14712105},
year = {2013},
date = {20130401},
urldate = {20211026},
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: prefixtrees and hashtables. 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: prefixtrees and hashtables. We propose a new approach to read alignment using random permutations of strings. 
Lederman, Roy R A Note about the ResolutionLength Characteristics of DNA Technical Report Yale CS (YALEU/DCS/TR1473), 2013. BibTeX  Tags: Sequence Alignment, Sequencing @techreport{lederman_note_2013,
title = {A Note about the ResolutionLength Characteristics of DNA},
author = {Roy R Lederman},
year = {2013},
date = {20130401},
number = {YALEU/DCS/TR1473},
pages = {6},
institution = {Yale CS},
keywords = {Sequence Alignment, Sequencing},
pubstate = {published},
tppubtype = {techreport}
}

Lederman, Roy R A permutationsbased algorithm for fast alignment of long pairedend reads Technical Report Yale CS (YALEU/DCS/TR1474), 2013. BibTeX  Tags: Algorithms, DNA sequencing, Fast algorithms, Randomized algorithms, Sequencing @techreport{lederman_permutationsbased_2013,
title = {A permutationsbased algorithm for fast alignment of long pairedend reads},
author = {Roy R Lederman},
year = {2013},
date = {20130401},
number = {YALEU/DCS/TR1474},
pages = {11},
institution = {Yale CS},
keywords = {Algorithms, DNA sequencing, Fast algorithms, Randomized algorithms, Sequencing},
pubstate = {published},
tppubtype = {techreport}
}

Lederman, Roy R Building approximate overlap graphs for DNA assembly using randompermutationsbased search Technical Report Yale CS (YALEU/DCS/TR1470), 2012. BibTeX  Tags: DNA sequencing, Sequencing @techreport{lederman_building_2012,
title = {Building approximate overlap graphs for DNA assembly using randompermutationsbased search},
author = {Roy R Lederman},
year = {2012},
date = {20121201},
number = {YALEU/DCS/TR1470},
pages = {10},
institution = {Yale CS},
keywords = {DNA sequencing, Sequencing},
pubstate = {published},
tppubtype = {techreport}
}

Lederman, Roy R Homopolymer Length Filters Technical Report Yale CS (YALEU/DCS/TR1465), 2012. BibTeX  Tags: Algorithms, DNA sequencing, Sequence Alignment, Sequencing @techreport{lederman_homopolymer_2012,
title = {Homopolymer Length Filters},
author = {Roy R Lederman},
year = {2012},
date = {20121001},
number = {YALEU/DCS/TR1465},
pages = {12},
institution = {Yale CS},
keywords = {Algorithms, DNA sequencing, Sequence Alignment, Sequencing},
pubstate = {published},
tppubtype = {techreport}
}
