**Welcome to Roy R. Lederman's homepage.**

I am 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.

### Interests and Recent work

**Mathematics of data science****The combination of inverse problems and unsupervised learning****Applied harmonics analysis****Numerical analysis and signal processing**: the truncated Fourier transform, prolate functions, the Laplace transform, decaying signals**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

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**.

**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

### 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

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, a method for recovering the common variable in multi-sensor experiments, is discussed in this paper, this technical report and this project webpage.

A different approach to the common variable recovery problem, which also constructs representations that are invariable to unknown transformations, is discussed in this paper.

** 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.

### Additional Computational Biology Algorithms

**Long-Range "Independence"**The repetitive nature of DNA strings is one of the challenges in read alignment. When one examines longer substrings of DNA, they appear less repetitive, or more unique; permutations-based algorithms benefit from this property. We describe a way of measuring the property in this paper and ways of using this property in reads with many "indels," in this paper.

**Homopolymer Length Filters**Homopolymer length filters eliminate the mapping problem caused by homopolymer length errors (ionTorrent/454). See paper.

**More information about my work in computational biology is available at**http://roy.lederman.name/compbio/ .

### Methodology

**Open-Source Proof**: experimental use of open-source code for analytical validation of expressions in papers. The expressions are generally obtained by other means, and we use Mathematica(tm) to validate the end results to the extent possible. Ultimately, the goal would be to have analytical code and numerical code validating the same expressions in a convenient manner, and Tex code produced directly from the analytical code. Preliminary examples can be found in the prolate project.

# Papers

Lederman, Roy R Numerical Algorithms for the Computation of Generalized Prolate Spheroidal Functions Paper 2017. Links | BibTeX | Tags: code, Cryo-EM, Generalized Prolated Spheroidal Functions (GPSF), Harmonic Analysis, Numerical Analysis, open-source, open-source paper, open-source proof, Prolate Spheroidal Wave Functions (PSWF), Signal Processing, SVD, Truncated Fourier Transform @misc{Lederman2017gpsf,
title = {Numerical Algorithms for the Computation of Generalized Prolate Spheroidal Functions}, author = {Roy R. Lederman}, url = {http://github.com/lederman/Prol https://arxiv.org/abs/1710.02874}, year = {2017}, date = {2017-10-08}, number = { arXiv:1710.02874}, keywords = {code, Cryo-EM, Generalized Prolated Spheroidal Functions (GPSF), Harmonic Analysis, Numerical Analysis, open-source, open-source paper, open-source proof, Prolate Spheroidal Wave Functions (PSWF), Signal Processing, SVD, Truncated Fourier Transform}, pubstate = {published}, tppubtype = {misc} } |

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. Links | BibTeX | Tags: Generalized Prolated Spheroidal Functions (GPSF), Harmonic Analysis, Laplace Transform, Numerical Analysis, Prolate Spheroidal Wave Functions (PSWF), Signal Processing, SVD, Truncated Fourier Transform, Truncated Laplace Transform @article{stability2017,
title = {Lower Bounds for Truncated Fourier and Laplace Transforms}, author = {Roy R. Lederman and Stefan Steinerberger}, url = {https://link.springer.com/article/10.1007/s00020-017-2364-z https://arxiv.org/abs/1605.03866}, year = {2017}, date = {2017-04-01}, journal = {Integral Equations and Operator Theory}, volume = {87}, number = {4}, pages = {529-543}, keywords = {Generalized Prolated Spheroidal Functions (GPSF), Harmonic Analysis, Laplace Transform, Numerical Analysis, Prolate Spheroidal Wave Functions (PSWF), Signal Processing, SVD, Truncated Fourier Transform, Truncated Laplace Transform}, pubstate = {published}, tppubtype = {article} } |

Lederman, Roy R; Rokhlin, Vladimir 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. Links | BibTeX | Tags: Harmonic Analysis, Laplace Transform, Numerical Analysis, Signal Processing, SVD, Truncated Laplace Transform @article{laplaceii2016,
title = {On the Analytical and Numerical Properties of the Truncated Laplace Transform - Part II}, author = {Roy R. Lederman and Vladimir Rokhlin}, url = {https://doi.org/10.1137/15M1028583}, doi = {10.1137/15M1028583}, year = {2016}, date = {2016-03-15}, journal = {SIAM Journal on Numerical Analysis}, volume = {54}, number = {2}, pages = {665–687}, keywords = {Harmonic Analysis, Laplace Transform, Numerical Analysis, Signal Processing, SVD, Truncated Laplace Transform}, pubstate = {published}, tppubtype = {article} } |

Lederman, Roy R; Rokhlin, Vladimir 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. Links | BibTeX | Tags: Harmonic Analysis, Laplace Transform, Numerical Analysis, Signal Processing, SVD, Truncated Laplace Transform @article{laplacei2015,
title = {On the Analytical and Numerical Properties of the Truncated Laplace Transform - I}, author = {Roy R. Lederman and Vladimir Rokhlin}, url = {https://doi.org/10.1137/140990681}, doi = {10.1137/140990681}, year = {2015}, date = {2015-05-12}, journal = {SIAM Journal on Numerical Analysis}, volume = {53}, number = {3}, pages = {1214-1235}, keywords = {Harmonic Analysis, Laplace Transform, Numerical Analysis, Signal Processing, SVD, Truncated Laplace Transform}, pubstate = {published}, tppubtype = {article} } |

Lederman, Roy R On the Analytical and Numerical Properties of the Truncated Laplace Transform PhD Thesis Yale University, 2014. Links | BibTeX | Tags: Harmonic Analysis, Laplace Transform, Numerical Analysis, Signal Processing, SVD, Truncated Laplace Transform @phdthesis{laplace2014,
title = {On the Analytical and Numerical Properties of the Truncated Laplace Transform}, author = {Roy R. Lederman}, url = {http://cpsc.yale.edu/sites/default/files/files/tr1490.pdf}, year = {2014}, date = {2014-05-00}, number = {1497}, institution = {YALE/DCS}, school = {Yale University}, keywords = {Harmonic Analysis, Laplace Transform, Numerical Analysis, Signal Processing, SVD, Truncated Laplace Transform}, pubstate = {published}, tppubtype = {phdthesis} } |

# Select Teaching

MATH555 / AMTH555 : Elements of Mathematical Machine Learning | Yale, Spring 2015 |

MATH 112 : Calculus of Functions of One Variable I | Yale, Spring 2015 |

AMTH 160 : The Structure of Networks – TA (Instructor: R.R. Coifman) | Yale, Spring 2014 |

AMTH 160 : The Structure of Networks – TA (Instructor: R.R. Coifman) | Yale, Spring 2013 |

AMTH 561 / CPSC 662 : Spectral Graph Theory – TA (Instructor: D.A. Spielman) | Yale, Fall 2012 |

CPSC 365 : Design and Analysis of Algorithms – TA (Instructor: D.A. Spielman) | Yale, Spring 2012 |

CPCS 445/545 : Introduction to Data Mining – TA (Instructor: V. Rokhlin) | Yale, Fall 2011 |