Georgia Institute of Technology College of Sciences

A fundamental problem in compressive sensing (CS) is to reconstruct a sparse signal under a few linear measurements far less than the physical dimension of the signal. Currently, CS favors incoherent systems, in which any two measurements are as little correlated as possible. In reality, however, many problems are coherent, in which case conventional methods, such as L1 minimization, do not work well. In this talk, I will present a novel non-convex approach, which is to minimize the difference of L1 and L2 norms, denoted as L1-L2, in order to promote sparsity. In addition to theoretical aspects of the L1-L2 approach, I will discuss two minimization algorithms. One is the difference of convex (DC) function methodology, and the other is based on a proximal operator, which makes some L1 algorithms (e.g. ADMM) applicable for L1-L2. Experiments demonstrate that L1-L2 improves L1 consistently and it outperforms Lp (p between 0 and 1) for highly coherent matrices. Some applications will be discussed, including super-resolution, image processing, and low-rank approximation.

Quantum theory includes many well-developed bounds for wave-functions, which can cast light on where they can be localized and where they are largely excluded by the tunneling effect. These include semiclassical estimates, especially the technique of Agmon, the use of "landscape functions," and some bounds from the theory of ordinary differential equations. With A. Maltsev of Queen Mary University I have been studying how these estimates of wave functions can be adapted to quantum graphs, which are by definition networks of one-dimensional Schrödinger equations joined at vertices.

Stochastic programming is concerned with decision making under uncertainty, seeking an optimal policy with respect to a set of possible future scenarios. While the value of Stochastic Programming is obvious to many practitioners, in reality uncertainty in decision making is oftentimes neglected. For deterministic optimisation problems, a coherent chain of modelling and solving exists. Employing standard modelling languages and solvers for stochastic programs is however difficult. First, they have (with exceptions) no native support to formulate Stochastic Programs. Secondly solving stochastic programs with standard solvers (e.g. MIP solvers) is often computationally intractable. David will be talking about his research that aims to make Stochastic Programming more accessible. First, he will be talking about modelling deterministic and stochastic programs in the Constraint Programming language MiniZinc - a modelling paradigm that retains the structure of a problem much more strongly than MIP formulations. Secondly, he will be talking about decomposition algorithms he has been working on to solve combinatorial Stochastic Programs.

Today's era of cloud computing is powered by massive data centers. A data center network enables the exchange of data in the form of packets among the servers within these data centers. Given the size of today's data centers, it is desirable to design low-complexity scheduling algorithms which result in a fixed average packet delay, independent of the size of the data center. We consider the scheduling problem in an input-queued switch, which is a good abstraction for a data center network. In particular, we study the queue length (equivalently, delay) behavior under the so-called MaxWeight scheduling algorithm, which has low computational complexity. Under various traffic patterns, we show that the algorithm achieves optimal scaling of the heavy-traffic scaled queue length with respect to the size of the switch. This settles one version of an open conjecture that has been a central question in the area of stochastic networks. We obtain this result by using a Lyapunov-type drift technique to characterize the heavy-traffic behavior of the expected total queue length in the network, in steady-state.