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Welcome to Roy R. Lederman’s homepage.

I am an Assistant Professor at the Department of Statistics and Data Science at Yale University. I am also a member of the Quantitative Biology Institute (QBio), the Applied Math Program, the Institute for Foundations of Data Science (FDS) and the Wu Tsai Institute (WTI) at Yale. I am a Sloan Research Fellow (2023).

In 2015-2018 I was a postdoc in the Program in Applied and Computational Mathematics at Princeton University, working with Amit Singer. In 2014-2015 I was a Gibbs Assistant Professor in the Applied Mathematics Program at Yale University, where I also got my PhD, working with Vladimir Rokhlin and Raphy Coifman. I have a BSc in physics and a BSc in electrical engineering from Tel-Aviv University.

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).


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Research

INTERESTS AND RECENT WORK

  • Mathematics of data science
  • The combination of inverse problems and unsupervised learning
  • Bayesian inference and variational inference
  • Numerical analysis and signal processing: the truncated Fourier transform, prolate functions, the Laplace transform, decaying signals
  • Applied harmonics analysis
  • Empirical geometry of data: unsupervised learning, manifold learning, diffusion maps, multi-sensor problems
  • 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CRYO-EM (a little out of date – more updates coming soon)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.

Recent recorded talks:
NYC Computational Cryo-EM Summer Workshop, August 2019
Computational Harmonic Analysis and Data Science, October 2019

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.


Preliminary results. See project page.

Acknowledgements: Adam Frost, Lakshmi Miller-Vedam, Joakim Anden


No, this is not a dancing cat. See project page.

BAYESIAN INFERENCE (more updates coming soon)

We are interested in revisiting elements of Bayesian Inference and Variational Inference. Some (but not all) of this work is motivated by applications such as cryo-EM (although it might not be obvious at first – for example questions about inference on manifolds). Example of some recent work in these areas: Evaluating the Implicit Midpoint Integrator for Riemannian Hamiltonian Monte Carlo (ICML) and Non-Canonical Hamiltonian Monte Carlo .

NUMERICAL ANALYSIS AND SIGNAL PROCESSING

Prolate Functions

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

Function06

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.

The numerical and analytical properties of the Truncated Laplace Transform are discussed in this paper (dissertation)this paper (part I) and this paper (part II).

Bounds on Transforms

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

GEOMETRY OF DATA

Alternating Diffusion Simulation

Alternating Diffusion, a method for recovering the common variable in multi-sensor experiments, is discussed in this paperthis 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.

What’s going on? Why is everything spinning? See project webpage,
this paper and in this report.

This experiment has nothing to do with the cryo-EM experiment above. Rotating animals are a very convenient visualization.
 

COMPUTATIONAL BIOLOGY

Random Permutations Based Alignment

I have developed randomized algorithms for sequencing of DNA and RNA.

Paper: “A Random-Permutations-Based Approach to Fast Read Alignment” (RECOMB-SEQ 2013). Also see this paper about the properties of DNA and sequencing.

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.

Publications

The most recent list can be found on on google scholar.

Herreros, D; Lederman, Roy R; Krieger, J; Jiménez-Moreno, 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

In: IUCrJ, vol. 8, no. 6, 2021, ISSN: 2052-2525, (Number: 6 Publisher: International Union of Crystallography).

Abstract | Links | BibTeX | Tags: cryo-EM, heterogeneity, Zernike

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

In: Microscopy and Microanalysis, vol. 27, no. S1, pp. 1680–1682, 2021, ISSN: 1431-9276, 1435-8115, (Publisher: Cambridge University Press).

Abstract | Links | BibTeX | Tags: cryo-EM, heterogeneity, Zernike

Brofos, James; Lederman, Roy R

Evaluating the Implicit Midpoint Integrator for Riemannian Hamiltonian Monte Carlo Inproceedings

In: International Conference on Machine Learning, pp. 1072–1081, PMLR, 2021, (ISSN: 2640-3498).

Links | BibTeX | Tags: HMC, Manifolds, MCMC, Numerical Analysis

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

Fan, Zhou; Lederman, Roy R; Sun, Yi; Wang, Tianhao; Xu, Sheng

Maximum likelihood for high-noise group orbit estimation and single-particle cryo-EM Technical Report

2021, (arXiv: 2107.01305).

Abstract | Links | BibTeX | Tags: Computer Science - Information Theory, Mathematics - Optimization and Control, Mathematics - Statistics Theory

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

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, Multi-view, multimodal, SPD Matrices, Statistics - Machine Learning

Lederman, Roy R; Andén, Joakim; Singer, Amit

Hyper-molecules: on the representation and recovery of dynamical structures for applications in flexible macro-molecules in cryo-EM Journal Article

In: Inverse Problems, vol. 36, no. 4, pp. 044005, 2020, ISSN: 0266-5611, 1361-6420.

Links | BibTeX | Tags: cryo-EM, heterogeneity, HyperMolecules, MCMC, Variational inference

Lederman, Roy R; Singer, Amit

A representation theory perspective on simultaneous alignment and classification Journal Article

In: Applied and Computational Harmonic Analysis, vol. 49, no. 3, pp. 1001–1024, 2020, ISSN: 1063-5203.

Abstract | Links | BibTeX | Tags: Algorithms, Alignment, Classification, cryo-EM, Graph-cut, heterogeneity, Heterogeneous multireference alignment, Representation Theory, Rotation group, SDP, Synchronization

Bandeira, Afonso S; Chen, Yutong; Lederman, Roy R; Singer, Amit

Non-unique games over compact groups and orientation estimation in cryo-EM Journal Article

In: Inverse Problems, vol. 36, no. 6, pp. 064002, 2020, ISSN: 0266-5611, 1361-6420.

Links | BibTeX | Tags: Algorithms, cryo-EM, Non-unique games, Representation Theory

Brofos, James A; Lederman, Roy R

Non-Canonical Hamiltonian Monte Carlo Technical Report

2020, (arXiv: 2008.08191).

Abstract | Links | BibTeX | Tags: Algorithms, Computer Science - Machine Learning, HMC, MCMC, Statistics - Machine Learning

Brofos, James A; Shu, Rui; Lederman, Roy R

A Bias-Variance Decomposition for Bayesian Deep Learning Inproceedings

In: pp. 14, 2019.

Abstract | BibTeX | Tags: Bayesian Deep Learning, Bayesian Inference, Deep Learning

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

Shnitzer, Tal; Lederman, Roy R; Liu, Gi-Ren; Talmon, Ronen; Wu, Hau-Tieng

Diffusion operators for multimodal data analysis Incollection

In: Handbook of Numerical Analysis, vol. 20, pp. 1–39, Elsevier, 2019, ISBN: 978-0-444-64140-3.

Links | BibTeX | Tags: Alternating Diffusion, BookChapter, Common variable, diffusion maps, Manifold Learning, Multi-view, multimodal, Multimodal data, Sensor fusion, Shape differences

Lederman, Roy R; Talmon, Ronen

Learning the geometry of common latent variables using alternating-diffusion Journal Article

In: Applied and Computational Harmonic Analysis, vol. 44, no. 3, pp. 509–536, 2018, ISSN: 1063-5203.

Abstract | Links | BibTeX | Tags: Algorithms, Alternating Diffusion, Alternating-diffusion, Common variable, diffusion maps, Diffusion-maps, Multi-view, multimodal, Multimodal analysis

Aldroubi, Akram; Huang, Longxiu; Krishtal, Ilya; Ledeczi, Akos; Lederman, Roy R; Volgyesi, Peter

Dynamical sampling with additive random noise Technical Report

no. arXiv:1807.10866 [math], 2018, (arXiv: 1807.10866).

Abstract | Links | BibTeX | Tags: Mathematics - Numerical Analysis

Shaham, Uri; Lederman, Roy R

Learning by coincidence: Siamese networks and common variable learning Journal Article

In: Pattern Recognition, vol. 74, pp. 52–63, 2018, ISSN: 00313203.

Links | BibTeX | Tags: Common variable, Deep Learning, Multi-view, multimodal, Siamese networks

Boumal, N; Bendory, T; Lederman, Roy R; Singer, A

Heterogeneous multireference alignment: A single pass approach Inproceedings

In: 2018 52nd Annual Conference on Information Sciences and Systems (CISS), pp. 1–6, 2018.

Abstract | Links | BibTeX | Tags: bispectrum, concave programming, cryo-EM, cyclic shifts, Discrete Fourier transforms, estimation theory, expectation-maximization, Gaussian mixture models, heterogeneity, heterogeneous MRA, Heterogeneous multireference alignment, Multireference alignment, Noise measurement, non-convex optimization, nonconvex optimization problem, Optimization, Reliability, signal estimation, signal processing, Signal resolution, Signal to noise ratio, single pass approach, Standards

Lederman, Roy R

Numerical Algorithms for the Computation of Generalized Prolate Spheroidal Functions Technical Report

2017.

Abstract | Links | BibTeX | Tags: Algorithms, cryo-EM, Fourier Transform, Numerical Analysis, Prolate, Slepian, Software

Lederman, Roy R; Steinerberger, Stefan

Lower Bounds for Truncated Fourier and Laplace Transforms Journal Article

In: Integral Equations and Operator Theory, vol. 87, no. 4, pp. 529–543, 2017, ISSN: 0378-620X, 1420-8989.

Links | BibTeX | Tags: Fourier Transform, Laplace Transform

Lederman, Roy R; Singer, Amit

Continuously heterogeneous hyper-objects in cryo-EM and 3-Đ movies of many temporal dimensions Technical Report

no. arXiv:1704.02899 [cs], 2017, (arXiv: 1704.02899).

Abstract | Links | BibTeX | Tags: Computer Science - Computer Vision and Pattern Recognition, cryo-EM, heterogeneity, HyperMolecules

Stanton, Kelly P; Jin, Jiaqi; Lederman, Roy R; Weissman, Sherman M; Kluger, Yuval

Ritornello: high fidelity control-free chromatin immunoprecipitation peak calling Journal Article

In: Nucleic Acids Research, vol. 45, no. 21, pp. e173–e173, 2017, ISSN: 0305-1048, (Publisher: Oxford Academic).

Abstract | Links | BibTeX | Tags: DNA sequencing, Sequencing, Software

Aldroubi, Akram; Huang, L; Krishtal, I; Lederman, Roy R

Dynamical sampling with random noise Inproceedings

In: 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

Lederman, Roy R; Singer, Amit

A Representation Theory Perspective on Simultaneous Alignment and Classification Technical Report

no. arXiv:1607.03464 [cs, math], 2016, (arXiv: 1607.03464).

Abstract | Links | BibTeX | Tags: Algorithms, Computer Science - Computer Vision and Pattern Recognition, cryo-EM, Mathematics - Optimization and Control, Representation Theory

Lederman, Roy R; Steinerberger, Stefan

Stability Estimates for Truncated Fourier and Laplace Transforms Technical Report

no. arXiv:1605.03866, 2016.

Abstract | Links | BibTeX | Tags: Laplace Transform

Lederman, Roy R; Rokhlin, V

On the Analytical and Numerical Properties of the Truncated Laplace Transform. Part II Journal Article

In: SIAM Journal on Numerical Analysis, vol. 54, no. 2, pp. 665–687, 2016, ISSN: 0036-1429, 1095-7170.

Links | BibTeX | Tags: Algorithms, Laplace Transform, Numerical Analysis

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, Multi-view

Lederman, Roy R; Talmon, Ronen; Wu, Hau-tieng; Lo, Yu-Lun; Coifman, Ronald R

Alternating diffusion for common manifold learning with application to sleep stage assessment Inproceedings

In: 2015 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), pp. 5758–5762, 2015, (ISSN: 2379-190X).

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

Lederman, Roy R; Rokhlin, V

On the Analytical and Numerical Properties of the Truncated Laplace Transform I. Journal Article

In: SIAM Journal on Numerical Analysis, vol. 53, no. 3, pp. 1214–1235, 2015, ISSN: 0036-1429, 1095-7170.

Links | BibTeX | Tags: Algorithms, Laplace Transform, Numerical Analysis

Lederman, Roy R; Talmon, Ronen

Common Manifold Learning Using Alternating-Diffusion Technical Report

Yale CS no. YALEU/DCS/TR-1497, 2014.

Links | BibTeX | Tags: AD, Algorithms, Alternating Diffusion, Manifold Learning

Lederman, Roy R

On the Analytical and Numerical Properties of the Truncated Laplace Transform PhD Thesis

Yale University, 2014, (YALEU/DCS/TR-1490).

BibTeX | Tags: Algorithms, Laplace Transform, Numerical Analysis

Lederman, Roy R

A random-permutations-based approach to fast read alignment Journal Article

In: BMC Bioinformatics, vol. 14, no. 5, pp. S8, 2013, ISSN: 1471-2105.

Abstract | Links | BibTeX | Tags: Algorithm, DNA sequencing, Fast algorithms, Neighbor Search, Random Permutation, Reference Genome, Reference Library, Search Problem

Lederman, Roy R

A Note about the Resolution-Length Characteristics of DNA Technical Report

Yale CS no. YALEU/DCS/TR-1473, 2013.

BibTeX | Tags: Sequence Alignment, Sequencing

Lederman, Roy R

A permutations-based algorithm for fast alignment of long paired-end reads Technical Report

Yale CS no. YALEU/DCS/TR-1474, 2013.

BibTeX | Tags: Algorithms, DNA sequencing, Fast algorithms, Randomized algorithms, Sequencing

Lederman, Roy R

Building approximate overlap graphs for DNA assembly using random-permutations-based search Technical Report

Yale CS no. YALEU/DCS/TR-1470, 2012.

BibTeX | Tags: DNA sequencing, Sequencing

Lederman, Roy R

Homopolymer Length Filters Technical Report

Yale CS no. YALEU/DCS/TR-1465, 2012.

BibTeX | Tags: Algorithms, DNA sequencing, Sequence Alignment, Sequencing

Teaching

S&DS262/S&DS562 : Computational Tools for Data ScienceYale, Fall 2021
S&DS262/S&DS562 : Computational Tools for Data ScienceYale, Spring 2021
S&DS262/S&DS562 : Computational Tools for Data ScienceYale, Spring 2020
S&DS663 : Computational Mathematics for Data ScienceYale, Fall 2019
S&DS676 : Signal Processing for Data ScienceYale, Spring 2019
S&DS663 : Computational Mathematics for Data ScienceYale, Fall 2018
MATH555 / AMTH555 : Elements of Mathematical Machine LearningYale, Spring 2015
MATH 112 : Calculus of Functions of One Variable IYale, Spring 2015