Group actions and symplectic techniques in Machine Learning and Computational Geometry

机器学习和计算几何中的群作用和辛技术

• 批准号: RGPIN-2017-06901
• 负责人: Fraser, Maia
• 金额: $ 3.64万
• 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: 2022
• 资助国家: 加拿大
• 起止时间: 2022-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=759203
• 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.

随着我们与世界的互动日益以数据为基础和相互联系，数学工具在广泛的科学学科中的相关性达到了前所未有的程度。拓扑学、几何学和统计学在计算机科学，特别是在数据科学中的地位日益上升，这一点尤其值得注意。数据是数字或基于数字的，因此通常带有潜在的拓扑、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将为算法开发带来立竿见影的好处，但这种创新需要计算机科学和数学方面的坚实基础。我的研究通过应用计算机科学中与群体行动和辛几何相关的数学工具，采取了一些措施来弥合这一差距。在这个建议中，我考虑了变换群以一种基本的方式作用于感兴趣对象的设置。这意味着我们知道感兴趣对象的G组对称性：例如，无论指纹如何旋转，扫描指纹都应该被认为是“相同的”-平面的旋转组就是对称性。数学中与研究群体行为相关的关键概念是不变性和等价性。利用先验的等方差可以定义更强大的代数不变量，但这在计算机科学中很少被利用。辛几何与群作用的研究密切相关，它是作为经典力学中运动方程的数学研究而出现的：一个系统在时间上以这样一种方式演化，即某些量是守恒的。接触几何是Pure Math的另一个密切相关的领域，有着相似的起源。这些几何领域很少在计算机科学中使用，但现在在机器学习中出现了很有前途的应用。我建议将与群动作和辛/接触几何相关的工具用于三个具体情况：(1)在机器学习中--分析由对称给出的底层结构如何能够提供更有效的学习策略，例如群不变特征选择，并使用辛-几何方法来设计算法；(2)在计算几何中--研究拓扑和群动作如何影响算法以及平面和曲面上的三角剖分的性质；(3)在接触几何中-利用等方差研究在某些接触流形上量子形式的柔性让位于经典形式的刚性的尺度的存在性。