Group actions and symplectic techniques in Machine Learning and Computational Geometry

机器学习和计算几何中的群行为和辛技术

• 批准号: RGPIN-2017-06901
• 负责人: Fraser, Maia
• 金额: $ 1.82万
• 依托单位: University of Ottawa
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2018
• 资助国家: 加拿大
• 起止时间: 2018-01-01 至 2019-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=651118
• 关键词: Group; actions; symplectic; techniques; Machine

Mathematical tools have become relevant across a broad range of scientific disciplines to an extent never before seen, as our interactions with the world become increasingly data-based and inter-connected. The rising role of topology, geometry and statistics in Computer Science, especially in Data Science, is particularly noticeable. Data are numbers or number-based and so often come with underlying topological, geometric or statistical structure. Understanding how this structure impacts algorithms and properties of interest confers an immediate benefit to algorithmic development, but this innovation requires solid foundations in both Computer Science and Mathematics. My research takes steps to bridge this gap, by applying mathematical tools related to group actions and symplectic geometry in Computer Science.******In this proposal, I consider settings where transformation groups act on objects of interest in an essential way. This means that we know a group G of symmetries of the objects of interest: for example a scan of an fingerprint should be considered "the same" regardless of how it is rotated - the group of rotations of the plane are the symmetries. Key notions from Math that are relevant in studying group actions are invariance and equivariance. Making use of equivariance a priori allows one to define more powerful algebraic invariants but this has rarely been leveraged in Computer Science. Closely linked with the study of group actions, symplectic geometry arose as the mathematical study of equations of motion in classical mechanics: a system evolves in time in such a way that certain quantities are conserved. Contact geometry is another closely related field of Pure Math with similar origins. These areas of geometry have rarely been used in Computer Science but promising applications are now appearing in Machine Learning.******I propose to bring tools related to group actions and symplectic/contact geometry to bear in three specific cases: (1) in Machine Learning - analyzing how underlying structure given by symmetries can inform more effective learning strategies, for example group-invariant feature selection, and using symplectic-geometric methods to design algorithms; (2) in Computational Geometry - investigating how topology and group actions affect algorithms and properties of triangulations in the plane and on surfaces; (3) in Contact Geometry - using equivariance to study the existence of a scale at which quantum-style flexibility gives way to classical-style rigidity in certain contact manifolds.

在从未见过的一定程度上，数学工具已经在广泛的科学学科中变得相关，因为我们与世界的互动变得越来越基于数据和相互联系。拓扑，几何学和统计学在计算机科学（尤其是数据科学）中的作用上升，特别明显。数据是基于数字或数字的，因此经常带有潜在的拓扑，几何或统计结构。了解这种结构如何影响算法和感兴趣的属性为算法开发带来了直接的好处，但是这种创新需要在计算机科学和数学中稳固的基础。我的研究采取了步骤来弥合这一差距，通过应用与计算机科学中的小组动作和符号几何形状相关的数学工具。******在此提案中，我考虑了转换组以一种基本方式对目标对象的设置。这意味着我们知道感兴趣对象的对称性的G组G组：例如，无论是如何旋转，指纹的扫描都应被视为“相同” - 平面的旋转组是对称性的。与研究小组行动相关的数学的关键概念是不变性和均衡。先验地利用均衡使人可以定义更强大的代数不变性，但这很少在计算机科学中利用。与对群体行为的研究密切相关，随着经典力学方程式的数学研究，出现了符号几何形状：系统以保守某些数量的方式随着时间的及时演变而来。接触几何是具有相似起源的纯数学的另一个密切相关的领域。这些几何领域很少在计算机科学中使用，但是现在在机器学习中出现了令人鼓舞的应用。 （2）在计算几何形状中 - 研究拓扑和群体作用如何影响平面和表面上三角形的算法和特性； （3）在接触几何形状中 - 使用均衡力研究量表的存在，在该量表中，量子式的柔韧性在某些接触歧管中取得了经典式刚性的范围。