[Aim of the Project]
We conducted a project "Application of Convex Programming to Statistical Science and Machine Learning." The project, which is collaboration among those who works in optimization and those who works in machine learning and statistics in the institute of statistical mathematics, aims at development of new models for statistical sciences and machine learning based on convex programming. Remarkable innovation of optimization algorithms in the last decade, in particular interior-point algorithms, enables us to solve new classes of convex programming problems such as convex quadratic programming problems, semidefinite programming problems and second-order cone programming problems. Development of support vector machine, a recent major breakthrough in machine learning, also seems to owe, to some extent, to progress of algorithms and software for convex quadratic programming. On the other hand, robust optimization, an area which becomes popular in optimization recently, shares a view with statistics and machine learning in that its main target is to develop methodologies handling uncertainty in the model and data to take the best strategy possible. Thus, there are nontrivial interactions between optimization and machine learning and statistical science. This is the motivation which leads us to this project.

[Activity of the Project]
During the period of this project, we had seminars to report progress of research of each member to exchange ideas and to read papers which would be interesting in the scope of the project. The topics included: density estimation based on semidefinite programming; optimization for dimension reduction in kernel methods; turbo code and optimization; theoretical estimation of performance of robust optimization based on sampling; estimation of function relation based on semidefinite programming; robust optimization of magnetic shielding and its performance analysis; clustering based on global optimization and semidefinite programming.
Here we pick up ﾔﾕRobust optimization of the magnetic shielding of the MAGLEV (MGEnetically LEViated train)ﾓ as an example of the research related to this project. MAGLEV train is held in the air and propelled with the strong magnetic field generated by superconducting magnetic unit. We need to enclose the car with shielding to protect passengers inside from the magnetic field outside. Design of the lightest shielding taking account of the magnetic field is formulated as a second-order cone programming problem. We developed a stochastic iterative method to obtain robust shielding taking account of uncertainty in the magnetic field outside. In the method, perturbed magnetic field is generated at each iteration and adaptation of the shielding to the perturbation is done by thickening it in such a way that the increase of the weight of the shielding is minimized using second-order cone programming. This process is regarded as a sort of learning. Performance of the designed robust shielding is evaluated with a statistical method based on the maximum likelihood method utilizing convex optimization.

Members
Takashi Tsuchiya