My new paper considers function interpolation via infinite-dimensional compressed sensing with weighted l1 minimization. In it I derive new recovery guarantees which are provably sharp in a number of important cases, and explain the practical usefulness of including weights in the optimization.
My summer undergraduate student Matthew King-Roskamp won second place in the poster competition at the 2015 SFU Symposium on Mathematics and Computation. Congratulations Matt!
I was awarded the AMMCS Kolmogorov-Wiener Prize for Young Researchers at the 2015 AMMCS-CAIMS Congress.
Rodrigo Platte and I have a new paper in which we propose a mapped polynomial method for high accuracy function approximation on arbitrary grids. We show how the method achieves numerical stability with near-optimal numbers of degrees of freedom – in effect circumventing the so-called impossibility theorem – by sacrificing classical convergence for finite, but arbitrarily high accuracy.
My new paper proposes an infinite-dimensional l1 minimization framework for function interpolation and analzes its approximation properties. In particular, I show that weighted l1 minimization with polynomials is an optimal method for approximating smooth, one-dimensional functions from scattered data.
I was awarded an Alfred P. Sloan Research Fellowship in Mathematics.
Siemens published practical confirmation of some of the ideas introduced in our paper concerning asymptotic sparsity, asymptotic incoherence and resolution dependence and their role in compressed sensing MRI. In particular, they concluded the following:
Current results practically demonstrated that it is possible to break the coherence barrier by increasing the spatial resolution in MR acquisitions. This likewise implies that the full potential of the compressed sensing is unleashed only if asymptotic sparsity and asymptotic incoherence is achieved.
Further information on these ideas, including the practical benefits they lead to, can be found in another recent work of ours here.