Briefly, my research is on mathematical signal processing using optimization.
These overview pages are updated too often, so see my recent papers to get the best idea of what I work on.
Topics: first-order methods, quasi-Newton methods, primal-dual algorithms
Optimization in the 21st century has changed dramatically due to the rapid increase in size of data sets. Interior-point methods, which were the ne plus ultra of the convex optimization community, cannot handle many of these data sets, and thus researchers turn to “older” methods such as first-order methods. The main challenges are handling non-smooth terms and constraints, and accelerating the convergence speed to levels comparable with limited-memory quasi-Newton methods (or at least Nesterov-accelerated methods).
Topics: compressed sensing and variants, matrix completion and variants (robust PCA…), non-negative matrix factorization and end-member detection, sparse SVM
Compressed sensing (CS) has taken the greater applied math community by storm since its introduction in 2004. Besides CS itself, the field has inspired new approaches to old problems in related disciplines. In particular, given the strengths of modern computers and algorithms, signal processing is increasingly relying on non-linear reconstruction algorithms (such as optimization-based algorithms). The main challenges are exploiting all prior information from the signal (going beyond just sparsity models), and solving optimization problems that are sometimes non-convex and usually large scale. The subfield of matrix completion is one exciting direction, and presents a plethora of optimization challenges due to the fundamental difference between matrix and vector optimization (i.e., linear programs vs semi-definite programs).
Topics: radar ADC using compressed sensing, quantum tomography, MRI, medical imaging, IMRT, renewable energy
No longer actively pursuing
I spent summer 2007 working for a tiny internet startup company “Taste Predictor” trying to win the Netflix prize and build collaborative filtering software. This was under the umbrella of ab inventio.
Liquid water can be supercooled to below the equilibrium freezing temperature. In this regime, many of water's unusual properties are amplified, and it also has some features, such as a liquid-liquid phase transition, that do not appear at higher temperatures. My undergraduate work on this was supervised by my thesis advisor Francis Starr of the Department of Physics at Wesleyan University. My undergraduate thesis was focused on the interplay of translational and rotational dynamics in supercooled water. Research was conducted by Molecular Dynamics simulation of ST2 water on computer clusters at both Wesleyan University in Connecticut and St. Francis Xavier University in Nova Scotia. Collaboration was also done with Peter Poole of St. Francis Xavier University.