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).

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CRYO-EM (a little out of date – more updates coming soon)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

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.

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

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.

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.

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.

@article{lederman_continuously_2017,
title = {Continuously heterogeneous hyper-objects in cryo-EM and 3-Đ movies of many temporal dimensions},
author = {Roy R Lederman and Amit Singer},
url = {http://arxiv.org/abs/1704.02899},
year = {2017},
date = {2017-04-01},
urldate = {2020-08-13},
journal = {arXiv:1704.02899 [cs]},
abstract = {Single particle cryo-electron microscopy (EM) is an increasingly popular method for determining the 3-D structure of macromolecules from noisy 2-D images of single macromolecules whose orientations and positions are random and unknown. One of the great opportunities in cryo-EM is to recover the structure of macromolecules in heterogeneous samples, where multiple types or multiple conformations are mixed together. Indeed, in recent years, many tools have been introduced for the analysis of multiple discrete classes of molecules mixed together in a cryo-EM experiment. However, many interesting structures have a continuum of conformations which do not fit discrete models nicely; the analysis of such continuously heterogeneous models has remained a more elusive goal. In this manuscript, we propose to represent heterogeneous molecules and similar structures as higher dimensional objects. We generalize the basic operations used in many existing reconstruction algorithms, making our approach generic in the sense that, in principle, existing algorithms can be adapted to reconstruct those higher dimensional objects. As proof of concept, we present a prototype of a new algorithm which we use to solve simulated reconstruction problems.},
note = {arXiv: 1704.02899},
keywords = {Computer Science - Computer Vision and Pattern Recognition, cryo-EM, heterogeneity, HyperMolecules},
pubstate = {published},
tppubtype = {article}
}

Single particle cryo-electron microscopy (EM) is an increasingly popular method for determining the 3-D structure of macromolecules from noisy 2-D images of single macromolecules whose orientations and positions are random and unknown. One of the great opportunities in cryo-EM is to recover the structure of macromolecules in heterogeneous samples, where multiple types or multiple conformations are mixed together. Indeed, in recent years, many tools have been introduced for the analysis of multiple discrete classes of molecules mixed together in a cryo-EM experiment. However, many interesting structures have a continuum of conformations which do not fit discrete models nicely; the analysis of such continuously heterogeneous models has remained a more elusive goal. In this manuscript, we propose to represent heterogeneous molecules and similar structures as higher dimensional objects. We generalize the basic operations used in many existing reconstruction algorithms, making our approach generic in the sense that, in principle, existing algorithms can be adapted to reconstruct those higher dimensional objects. As proof of concept, we present a prototype of a new algorithm which we use to solve simulated reconstruction problems.

@article{lederman_stability_2016,
title = {Stability Estimates for Truncated Fourier and Laplace Transforms},
author = {Roy R Lederman and Stefan Steinerberger},
url = {https://arxiv.org/abs/1605.03866v1},
year = {2016},
date = {2016-05-01},
urldate = {2020-08-13},
abstract = {We prove sharp stability estimates for the Truncated Laplace Transform and
Truncated Fourier Transform. The argument combines an approach recently
introduced by Alaifari, Pierce and the second author for the truncated Hilbert
transform with classical results of Bertero, Grünbaum, Landau, Pollak and
Slepian. In particular, we prove there is a universal constant $c textgreater0$ such that
for all $f textbackslashin Ltextasciicircum2(textbackslashmathbbR)$ with compact support in $[-1,1]$ normalized to $textbackslashtextbarftextbackslashtextbar_Ltextasciicircum2[-1,1] = 1$ $$ textbackslashint_-1textasciicircum1textbartextbackslashwidehatf(ξ)textbartextasciicircum2dξ textbackslashgtrsim
textbackslashleft(ctextbackslashlefttextbackslashtextbarf_x textbackslashrighttextbackslashtextbar_Ltextasciicircum2[-1,1] textbackslashright)textasciicircumvphantom- ctextbackslashlefttextbackslashtextbarf_x
textbackslashrighttextbackslashtextbar_Ltextasciicircum2[-1,1]vphantom$$ The inequality is sharp in the sense that there is an
infinite sequence of orthonormal counterexamples if $c$ is chosen too small.
The question whether and to which extent similar inequalities hold for generic
families of integral operators remains open.},
keywords = {Laplace Transform},
pubstate = {published},
tppubtype = {article}
}

We prove sharp stability estimates for the Truncated Laplace Transform and
Truncated Fourier Transform. The argument combines an approach recently
introduced by Alaifari, Pierce and the second author for the truncated Hilbert
transform with classical results of Bertero, Grünbaum, Landau, Pollak and
Slepian. In particular, we prove there is a universal constant $c textgreater0$ such that
for all $f textbackslashin Ltextasciicircum2(textbackslashmathbbR)$ with compact support in $[-1,1]$ normalized to $textbackslashtextbarftextbackslashtextbar_Ltextasciicircum2[-1,1] = 1$ $$ textbackslashint_-1textasciicircum1textbartextbackslashwidehatf(ξ)textbartextasciicircum2dξ textbackslashgtrsim
textbackslashleft(ctextbackslashlefttextbackslashtextbarf_x textbackslashrighttextbackslashtextbar_Ltextasciicircum2[-1,1] textbackslashright)textasciicircumvphantom- ctextbackslashlefttextbackslashtextbarf_x
textbackslashrighttextbackslashtextbar_Ltextasciicircum2[-1,1]vphantom$$ The inequality is sharp in the sense that there is an
infinite sequence of orthonormal counterexamples if $c$ is chosen too small.
The question whether and to which extent similar inequalities hold for generic
families of integral operators remains open.

@techreport{shaham_common_2015,
title = {Common Variable Learning and Invariant Representation Learning using Siamese Neural Networks},
author = {Uri Shaham and Roy R Lederman},
url = {https://arxiv.org/abs/1512.08806v3},
year = {2015},
date = {2015-12-01},
urldate = {2020-08-13},
abstract = {We consider the statistical problem of learning common source of variability
in data which are synchronously captured by multiple sensors, and demonstrate
that Siamese neural networks can be naturally applied to this problem. This
approach is useful in particular in exploratory, data-driven applications,
where neither a model nor label information is available. In recent years, many
researchers have successfully applied Siamese neural networks to obtain an
embedding of data which corresponds to a "semantic similarity". We present an
interpretation of this "semantic similarity" as learning of equivalence
classes. We discuss properties of the embedding obtained by Siamese networks
and provide empirical results that demonstrate the ability of Siamese networks
to learn common variability.},
keywords = {Common variable, Deep Learning, Multi-view},
pubstate = {published},
tppubtype = {techreport}
}

We consider the statistical problem of learning common source of variability
in data which are synchronously captured by multiple sensors, and demonstrate
that Siamese neural networks can be naturally applied to this problem. This
approach is useful in particular in exploratory, data-driven applications,
where neither a model nor label information is available. In recent years, many
researchers have successfully applied Siamese neural networks to obtain an
embedding of data which corresponds to a "semantic similarity". We present an
interpretation of this "semantic similarity" as learning of equivalence
classes. We discuss properties of the embedding obtained by Siamese networks
and provide empirical results that demonstrate the ability of Siamese networks
to learn common variability.

@techreport{lederman_permutations-based_2013,
title = {A permutations-based algorithm for fast alignment of long paired-end reads},
author = {Roy R Lederman},
year = {2013},
date = {2013-04-01},
number = {YALEU/DCS/TR-1474},
pages = {11},
institution = {Yale CS},
keywords = {Algorithms, DNA sequencing, Fast algorithms, Randomized algorithms, Sequencing},
pubstate = {published},
tppubtype = {techreport}
}