Tensor Decomposition - an algebraic and geometric approach

张量分解 - 一种代数和几何方法

• 批准号: RGPIN-2016-04683
• 负责人: Geramita, Anthony
• 金额: $ 1.4万
• 依托单位: Queen's University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2016
• 资助国家: 加拿大
• 起止时间: 2016-01-01 至 2017-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=615239
• 关键词: Tensor; Decomposition; algebraic; geometric; approach

The proposal for the research described concerns problems in the general area of tensor decomposition i.e. for a given tensor one is asked to find the minimal way to represent that tensor as a sum, using members of some family of simpler tensors. This elementary sounding statement actually contains many interesting and perplexing problems, not only in mathematics, but in physics, biology, statistics, computational complexity theory and communication theory. The wide range of applications for solutions to these problems has not only stimulated mathematicians to find solutions, but the applications themselves have suggested new and interesting mathematical approaches to these problems. This cross fertilization makes the area a rich and interesting place to put one's effort and encourages cooperation between scientists who are not always used to having important things to say to each other. To have e.g., statisticians, algebraic geometers and engineers working on the same problems is, to me, very exciting. It is anticipated that the research results in this area will increase our understanding of these fundamental problems and provide profound insight into their very nature. Coming at these problems from a variety of directions cannot but help us find new ways to think about what the solutions should be like. The variety of the approaches and the vastness of the problems makes this also an excellent area for researchers of every level to study. As I mention in the proposal, there is room for undergraduates, doing experiments with the problems, to Postdoctoral fellows, searching for patterns in the nature of the solutions. As for the impact that this work might have in Canada, I believe, that like most of mathematical and scientific research, the impact is worldwide since the audience does not have national boundaries. That being said, if solutions to these problems are found in Canada this will reflect very positively on our stature as a place to do research and to work with people who are finding answers to important questions. When a Canadian wins a Nobel Prize, the results of that research do not only help Canadians but the results do say that Canada is a place where excellent work is being done.

该建议的研究所描述的问题，在一般领域的张量分解，即对于一个给定的张量之一，被要求找到最小的方式来表示张量作为一个总和，使用一些家庭的成员更简单的张量。这个听起来很简单的陈述实际上包含了许多有趣和令人困惑的问题，不仅在数学中，而且在物理学，生物学，统计学，计算复杂性理论和通信理论中。 这些问题的解决方案的广泛应用不仅刺激了数学家找到解决方案，但应用程序本身提出了新的和有趣的数学方法来解决这些问题。这种交叉施肥使该地区成为一个丰富而有趣的地方，可以投入精力，并鼓励科学家之间的合作，这些科学家并不总是习惯于彼此有重要的事情要说。例如，统计学家、代数几何学家和工程师一起研究同样的问题，对我来说，非常令人兴奋。 预计这一领域的研究成果将增加我们对这些基本问题的理解，并对其本质提供深刻的见解。从不同的方向来看待这些问题，只能帮助我们找到新的方法来思考解决方案应该是什么样的。方法的多样性和问题的广泛性使得这也是各个层次研究人员研究的一个很好的领域。正如我在提案中提到的，有本科生的空间，做实验的问题，博士后研究员，寻找模式的性质的解决方案。 至于这项工作可能在加拿大产生的影响，我相信，像大多数数学和科学研究一样，影响是世界性的，因为观众没有国界。话虽如此，如果在加拿大找到这些问题的解决方案，这将非常积极地反映我们作为进行研究并与寻找重要问题答案的人们合作的地方的地位。当一个加拿大人获得诺贝尔奖时，这项研究的结果不仅帮助了加拿大人，而且还表明加拿大是一个正在进行出色工作的地方。