报告题目：On decompositions and approximations of conjugate partial-symmetric complex tensors

报告人：江波副教授（博导、上海财经大学）

报告时间：2018年11月12日周一上午10:00

报告地点：数计学院4号楼229报告厅

报告摘要：Conjugate partial-symmetric (CPS) tensors are the high-order generalization of Hermitian matrices. As the role played by Hermitian matrices in matrix theory and quadratic optimization, CPS tensors have shown growing interest recently in tensor theory and optimization, particularly in many application-driven complex polynomial optimization problems. In this paper, we study CPS tensors with a focus on ranks, rank-one decompositions and approximations, as well as their applications. The analysis is conducted along side with a more general class of complex tensors called partial-symmetric tensors. We prove constructively that any CPS tensor can be decomposed into a sum of rank-one CPS tensors, which provides an alternative definition of CPS tensors via linear combinations of rank-one CPS tensors. Three types of ranks for CPS tensors are defined and shown to be different in general. This leads to the invalidity of the conjugate version of Comon's conjecture. We then study rank-one approximations and matricizations of CPS tensors. By carefully unfolding CPS tensors to Hermitian matrices, rank-one equivalence can be preserved. This enables us to develop new convex optimization models and algorithms to compute best rank-one approximation of CPS tensors. Numerical experiments from various data are performed to justify the capability of our methods.

报告人简介：江波，上海财经大学博导，副教授，于2013年9月在美国明尼苏达大学工业与系统工程系获得博士学位。主要研究领域包括优化理论，收益管理，信号处理等。在多项式优化近似算法，低秩张量优化，非负张量的表达以及共轭复张量的表达等方向取得了一系列重要结果。相关结果发表在了《Mathematics of Operations Research》，《Mathematical Programming》，《SIAM Journal on Matrix Analysis and Applications》，《Foundations of Computational Mathematics》等运筹优化领域的国际著名杂志上。曾在美国对冲基金公司Whitebox Advisors担任暑期研究员，从事鲁棒组合投资的研究工作。现为美国数学会旗下 Mathematical Reviews 的评论员，主持过多项国家自然科学基金，荣获2015年上海财经大学学术新人奖，2016年申万宏源奖教金。