Wotao's research summary

Sparse optimization

What is common in all l1 or l1-like minimization problems mentioned above is that the data is dense; it is the solutions that are sparse. Traditional approaches for solving these problems reduce them to standard optimization problems, which are then solved by algorithms that exploit only data structures such as matrix sparsity. Therefore, the challenge is to develop algorithms that take advantages of the structure of the solutions rather that of the data. To address this challenge, I have been recently working on first-order algorithms specifically designed for solving large-scale optimization problems involving various l1 and l1-like functions. I also worked on new total variation algorithms that yields recovered images of screen resolutions as fast as in seconds. The underlying idea also led to fast algorithms for color image processing and MR image reconstruction. In addition, for iterative thresholding based algorithms, we discovered useful properties such as quick identification of nonzero entries in sparse solutions and the q-linear global convergence. These results are potentially useful in reducing the original problem, before it is fully solved, into one that is much smaller. It was also discovered that the Basis Pursuit problem, a constrained optimization for signal recovery in compressed sensing, can be solved by the Bregmen method, which takes only a small and provably finite number of iterations. A slightly different version of it, called the linearized Bregman method, is more interesting because its intermediate solutions are mostly sparse, making the method especially suitable for certain matrix problems. In this research I expect to develop a set of strategies of exploiting solution structures that can be widely applied to many algorithms.

Compressed sensing

Recent developments in mathematics and information theory have led to the emergence of compressed sensing, which acquires a digital representation of an object of interest, not by taking traditional samples, but by measuring a rather small number of linear projections followed by solving an optimization problem to reconstruct the object. The reconstruction problem can be viewed as an inverse problem; in particular, it uses l1 or l1-like (e.g., total variation) regularization to obtain sparse solutions. Based my existing work on compressed sensing, I intend to develop and implement new algorithms for faster and more stable signal recovery, study how to take advantage of prior knowledge including signal structures and distributions of nonzero values, extend both the methodology and the algorithms to other problems such matrix rank minimization, as well as apply compressed sensing to solving PDE-based inverse problems with solutions being sparse in a way.

Ill-posed inverse problems

Ill-posedness refers to the nature of certain inverse problems that either there exist more than one solution or small noise in the data may result in large errors in the solutions; hence regularization is introduced to obtain a unique, desired solution and control the errors. I have worked extensively on regularization techniques to improve solution quality in inverse problems. For example, Bregman iterative regularization (which is slightly different from the Bregman method above) was introduced to yield restored images that are sharper and have higher contrast. To reconstruct MR images with more small-scale details, total variation is combined with the l1-norm of wavelets coefficients in a compressed sensing recovery problem. In my plan, I will continue the research on regularization and, in particular, study approaches such as parallel imaging, multi-scale methods, as well as those based on the so-called nonlocal information.