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CRYO-EM (a little out of date – more updates coming soon)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

@techreport{brofos_magnetic_2020,
title = {Magnetic Manifold Hamiltonian Monte Carlo},
author = {James A Brofos and Roy R Lederman},
url = {http://arxiv.org/abs/2010.07753},
year = {2020},
date = {2020-10-01},
urldate = {2020-11-25},
abstract = {Markov chain Monte Carlo (MCMC) algorithms offer various strategies for sampling; the Hamiltonian Monte Carlo (HMC) family of samplers are MCMC algorithms which often exhibit improved mixing properties. The recently introduced magnetic HMC, a generalization of HMC motivated by the physics of particles influenced by magnetic field forces, has been demonstrated to improve the performance of HMC. In many applications, one wishes to sample from a distribution restricted to a constrained set, often manifested as an embedded manifold (for example, the surface of a sphere). We introduce magnetic manifold HMC, an HMC algorithm on embedded manifolds motivated by the physics of particles constrained to a manifold and moving under magnetic field forces. We discuss the theoretical properties of magnetic Hamiltonian dynamics on manifolds, and introduce a reversible and symplectic integrator for the HMC updates. We demonstrate that magnetic manifold HMC produces favorable sampling behaviors relative to the canonical variant of manifold-constrained HMC.},
note = {arXiv: 2010.07753},
keywords = {Algorithms, Computer Science - Machine Learning, HMC, Manifolds, MCMC, Statistics - Machine Learning},
pubstate = {published},
tppubtype = {techreport}
}

Markov chain Monte Carlo (MCMC) algorithms offer various strategies for sampling; the Hamiltonian Monte Carlo (HMC) family of samplers are MCMC algorithms which often exhibit improved mixing properties. The recently introduced magnetic HMC, a generalization of HMC motivated by the physics of particles influenced by magnetic field forces, has been demonstrated to improve the performance of HMC. In many applications, one wishes to sample from a distribution restricted to a constrained set, often manifested as an embedded manifold (for example, the surface of a sphere). We introduce magnetic manifold HMC, an HMC algorithm on embedded manifolds motivated by the physics of particles constrained to a manifold and moving under magnetic field forces. We discuss the theoretical properties of magnetic Hamiltonian dynamics on manifolds, and introduce a reversible and symplectic integrator for the HMC updates. We demonstrate that magnetic manifold HMC produces favorable sampling behaviors relative to the canonical variant of manifold-constrained HMC.

@inproceedings{lederman_alternating_2015,
title = {Alternating diffusion for common manifold learning with application to sleep stage assessment},
author = {Roy R Lederman and Ronen Talmon and Hau-tieng Wu and Yu-Lun Lo and Ronald R Coifman},
doi = {10.1109/ICASSP.2015.7179075},
year = {2015},
date = {2015-01-01},
booktitle = {2015 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)},
pages = {5758--5762},
abstract = {In this paper, we address the problem of multimodal signal processing and present a manifold learning method to extract the common source of variability from multiple measurements. This method is based on alternating-diffusion and is particularly adapted to time series. We show that the common source of variability is extracted from multiple sensors as if it were the only source of variability, extracted by a standard manifold learning method from a single sensor, without the influence of the sensor-specific variables. In addition, we present application to sleep stage assessment. We demonstrate that, indeed, through alternating-diffusion, the sleep information hidden inside multimodal respiratory signals can be better captured compared to single-modal methods.},
note = {ISSN: 2379-190X},
keywords = {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},
pubstate = {published},
tppubtype = {inproceedings}
}

In this paper, we address the problem of multimodal signal processing and present a manifold learning method to extract the common source of variability from multiple measurements. This method is based on alternating-diffusion and is particularly adapted to time series. We show that the common source of variability is extracted from multiple sensors as if it were the only source of variability, extracted by a standard manifold learning method from a single sensor, without the influence of the sensor-specific variables. In addition, we present application to sleep stage assessment. We demonstrate that, indeed, through alternating-diffusion, the sleep information hidden inside multimodal respiratory signals can be better captured compared to single-modal methods.