# Home

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.

Between 1/2016 and 3/2016 I will be at the Hausdorff Research Institute for Mathematics (HIM) in Bonn for the Mathematics of Signal Processing trimester program.

In 2014-2015 I was a Gibbs Assistant Professor in the Applied Mathematics Program at Yale University, working with Vladimir Rokhlin and Raphy Coifman.

I organize the IDeAS seminar at Princeton, please email me if you would like to give a talk.

# Research

Some Recent work:

• Numerical analysis: signal processing, the Laplace transform, decaying signals.
• Empirical geometry of data: manifold learning, diffusion maps, multi-sensor problems, unordered datasets.
• Computational biology: fast search algorithms, statistics of DNA, sequencing, organization of biological data.

## Numerical Analysis and Signal Processing

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:
$\left({L}_{a,b}(f)\right)(s) = \int_a^b e^{-st}f(t) dt$ .

The numerical and analytical properties of the Truncated Laplace Transform are discussed here.

## Geometry of Data

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

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

## 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 [Charikar, 2002] (random permutations in search without the special properties for the sequencing problem), and this report on the properties of sequencing.

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: technical report.

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 report and ways of using this property in reads with many “indels,” in this report.

Homopolymer Length Filters
Homopolymer length filters eliminate the mapping problem caused by homopolymer length errors (ionTorrent/454). A technical report is available here.

# Papers and Technical Reports

 (In preparation) On the Analytical and Numerical Properties of the Truncated Laplace Transform III. Lederman, R. R. and Rokhlin, V. (In Publication) On the Analytical and Numerical Properties of the Truncated Laplace Transform II. Shaham, U. and Lederman, R. R. (2015) Common Variable Discovery and Invariant Representation Learning using Artificial Neural Networks. Technical Report. Lederman, R. R. and Rokhlin, V. (2015). On the Analytical and Numerical Properties of the Truncated Laplace Transform I. SIAM Journal on Numerical Analysis, 53(3), 1214-1235. Lederman, R. R., Talmon, R., Wu, H., Lo, Y. and Coifman, R.R. (2015) Alternating Diffusion for Common Manifold Learning with Application to Sleep Stage Assessment. ICCASP. Lederman, R. R. and Talmon, R. (In publication, technical report) Learning the geometry of common latent variables using alternating-diffusion. Lederman, R. R. (2014) On the Analytical and Numerical Properties of the Truncated Laplace Transform. (Dissertation) Lederman, R. R. (2013) A Random-Permutations-Based Approach to Fast Read Alignment. BMC Bioinformatics 2013, 14(Suppl 5):S8 Lederman, R. R. (2013) Using the Long Range “Independence” in DNA: Coupled-Seeds and Pre-Alignment Filters. Technical Report. Lederman, R. R. (2013) A Permutations-Based Algorithm for Fast Alignment of Long Paired-End Reads. Technical Report. Lederman, R. R. (2012) A Note About the Resolution-Length Characteristics of DNA. Technical Report. Lederman, R. R. (2012) Building Approximate Overlap Graphs for DNA Assembly Using Random-Permutations-Based Search. Technical Report. Lederman, R. R. (2012) Homopolymer Length Filters. Technical Report.

# Select Talks and Posters

 Talk SIAM CSE 2015 Salt Lake City, UT 2015 Common Manifold Learning Using Alternating Diffusion for Multimodal Signal Processing. Talk IDeAS seminar, Princeton Princeton, NJ 2014 Common Manifold Learning Using Alternating-Diffusion. Talk ICERM Providence, RI 2014 Algorithms for DNA Sequencing. Poster ICERM Providence, RI 2014 SVD of the truncated Laplace transform. Talk Seminar, Weizmann Institute Rehovot, Israel 2014 On the Analytical and Numerical Properties of the Truncated Laplace Transform. Talk Applied Mathematics Seminar, Tel Aviv University Tel Aviv, Israel 2014 On the Analytical and Numerical Properties of the Truncated Laplace Transform. Talk Yale Applied Mathematics Seminar New Haven, CT 2013 Algorithms for DNA Sequencing. Posters Genome Informatics conference CSHL, NY 2013 Using the Long-Range “independence” Property of DNA for Read Mapping. General Purpose and Customized Random-Permutations-Based Mappers. Talk Broad Inst. of MIT and Harvard Cambridge, MA 2013 New Randomized Approaches to Fast and Accurate Read Processing, Mapping and Assembly. Talk, Poster Recomb and Recomb-Seq Conference Beijing, China 2013 A Random-Permutations-Based Approach to Fast Read Alignment. Poster Big Data Science Symposium New Haven, CT 2012 Fast Processing of DNA Sequences: Random-Permutations and HPL Filters. Poster HitSeq and ISMB conference Long Beach, CA 2012 “Shuffling-Based” Fast Read Alignment. Poster CJR Conference New Haven, CT 2012 “Random-Projections-Like” DNA Read Alignment.

# 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