Group actions and symplectic techniques in Machine Learning and Computational Geometry

随着我们与世界的互动越来越多地以数据为基础和相互联系，数学工具在广泛的科学学科中发挥了前所未有的作用。拓扑学、几何学和统计学在计算机科学，尤其是数据科学中的作用日益突出。数据是数字或基于数字的，因此通常具有潜在的拓扑结构、几何结构或统计结构。了解这种结构如何影响算法和感兴趣的属性，可以为算法的发展带来直接的好处，但这种创新需要在计算机科学和数学方面打下坚实的基础。我的研究通过应用与计算机科学中的群体行为和辛几何相关的数学工具来弥合这一差距。******在这个建议中，我考虑了转换组以基本方式作用于感兴趣的对象的设置。这意味着我们知道感兴趣对象的G组对称性：例如，无论如何旋转，指纹的扫描都应该被认为是“相同的”——平面的旋转组是对称的。数学中与研究群体行为相关的关键概念是不变性和等变性。利用先验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。辛几何与群体行为的研究密切相关，它是作为经典力学中运动方程的数学研究而出现的：一个系统随着时间的推移而演变，以至于某些量是守恒的。接触几何是另一个与纯数学密切相关的领域，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有前途的应用。******我建议在三个具体案例中引入与群体行为和辛/接触几何相关的工具：(1)在机器学习中-分析对称给出的底层结构如何能够告知更有效的学习策略，例如群不变特征选择，并使用辛几何方法来设计算法；(2)计算几何-研究拓扑和群体行为如何影响平面和表面上三角形的算法和性质；(3)在接触几何中-使用等方差研究在某些接触流形中量子型柔性让位给经典型刚性的尺度的存在性。

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.