**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.

### Some Recent work

**Empirical geometry of data**: manifold learning, diffusion maps, multi-sensor problems, unordered datasets**Cryo-EM**: application 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**Numerical analysis**: signal processing, the Laplace transform, decaying signals

### Cryo-EM

Cryo-electron microscopy (cryo-EM) is a method for imaging molecules without crystallization. It has been named “Method of the Year 2015” by Nature Methods.

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 report**.

I also work on a representation theory approach to simultaneous alignment and classification (this work is not related to the movie below), with applications in the heterogeneity problem in cryo-EM.

**For more information on my work in cryo-EM, see project page.**

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

### 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.

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

### 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 technical report.

** 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 report on the properties of 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: technical report.

### 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 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.

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

# Papers and Technical Reports

(In preparation) On the Analytical and Numerical Properties of the Truncated Laplace Transform III. |

Lederman, R. R., and Singer, A. (2017) Continuously heterogeneous hyper-objects in cryo-EM and 3-D movies of many temporal dimensions. arXiv preprint arXiv:1704.02899. |

Lederman, R. R., and Steinerberger, S., (2017) Lower Bounds for Truncated Fourier and Laplace Transforms. Integr. Equ. Oper. Theory 87: 529. doi:10.1007/s00020-017-2364-z . |

Lederman, R. R., and Steinerberger, S., (2016) Stability Estimates for Truncated Fourier and Laplace Transforms. arXiv preprint arXiv:1605.03866 . |

Lederman, R. R., and Singer, A. (2016) A Representation Theory Perspective on Simultaneous Alignment and Classification. arXiv preprint arXiv:1607.03464. |

Lederman, R. R. and Rokhlin, V. (2016) On the Analytical and Numerical Properties of the Truncated Laplace Transform. Part II. SIAM Journal on Numerical Analysis 54.2: 665-687. |

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. (2015) Learning the geometry of common latent variables using alternating-diffusion. Applied and Computational Harmonic Analysis. |

Lederman, R. R. and Talmon, R. (2015), (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 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 |