PIMS Lunchbox Lecture: Dr. Ming-Jun Lai

I shall first explain the concept of sparse solution of rectangular linear system. Algorithms such as L1 minimization and greedy strategy will be introduced for finding a sparse solution of underdetermined linear system. As an example, economic representation of numerical solution to the Poisson equation based on nested spline subspaces will be presented. Next, I shall explain the problem of matrix completion. Several algorithms for low rank matrix completion will be outlined together with convergence analysis. An example of removing wires from an image will be demonstrated. Then I shall explain the sparse phase retrieval problem which is to recover the signals from the magnitude of the measurements. Traditionally, one solves a nonlinear least squares problem. I will show how to solve it using a minimization with minimizing functional consisting of a difference of two convex functions. Finally, I shall explain how to solve the graph clustering problem by a sparse solution technique. Examples of sorting people from their shopping lists will be demonstrated.