Group actions and symplectic techniques in Machine Learning and Computational Geometry

点击翻译按钮获取中文摘要

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.