INTERESTS AND RECENT WORK
- Mathematics of data science
- The combination of inverse problems and unsupervised learning
- Bayesian inference and variational inference
- Numerical analysis and signal processing: the truncated Fourier transform, prolate functions, the Laplace transform, decaying signals
- Applied harmonics analysis
- 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CRYO-EM (a little out of date – more updates coming soon)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.
Preliminary results. See project page.
Acknowledgements: Adam Frost, Lakshmi Miller-Vedam, Joakim Anden
No, this is not a dancing cat. See project page.
BAYESIAN INFERENCE (more updates coming soon)
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
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.
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.
This experiment has nothing to do with the cryo-EM experiment above. Rotating animals are a very convenient visualization.
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.