| Optimization over Orbit Closures of Real Reductive Lie Group Actions |
 Zhan Zhiyuan
（指導教員：平井 広志 准教授／数理情報第2研究室）
Given a nonnegative square matrix A，the matrix scaling problem is determine if there is a scaling B of A, that is B = XAY where X,Y are diagonal matrices, such that B is an approximately double stochastic matrix. This problem is connected with questions arising from many areas. For example, for the matrix A, it can construct a bipartite graph G, like the figure show. Then A is approximately doubly stochastic scalable if and only if the corresponding graph has a perfect matching. In fact, the matrix scaling problem is a special case of a classical problem in invariant theory, which can be viewed as optimizing a function. In this thesis, we considered subgroups in real general linear group acting on real vector spaces. Then this problem can be generalized to optimize the Kempf-Ness function on a Riemannian manifold. So, we applied the Riemannian gradient descent algorithm to this problem.
During these two years, I have learned a lot of knowledge from different areas and accumulated precious experiences in my research. I would like to express my appreciation to my supervisor, Prof. Hiroshi Hirai and Prof. Kunihiko Sadakane, for their instructions. And I am grateful to members in 2nd laboratory for their help in my life.
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