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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) at Yale.

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CRYO-EM (a little out of date – more updates coming soon)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

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

Diffusion operators for multimodal data analysis Incollection

Handbook of Numerical Analysis, 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

Applied and Computational Harmonic Analysis, 44 (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, test

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

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

Teaching

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