Geometry of Data
Topological data analysis asks when balls in a metric space $(X,d)$ intersect. Geometric data analysis asks how much balls have to be enlarged to intersect. We connect this principle to the traditional core geometric concept of curvature. This enables us, on one hand, to reconceptualize curvature and link it to the geometric notion of hyperconvexity. On the other hand, we can then also understand methods of topological data analysis from a geometric perspective.
Code (0)
등록된 구현이 없습니다.
Tasks
Topological Data AnalysisSimilar Papers 제목 키워드 기반
Gold-medalist Performance in Solving Olympiad Geometry with AlphaGeometry2
We present AlphaGeometry2, a significantly improved version of AlphaGeometry introduced in Trinh et al. (2024), which has now surpassed an average gold medalist in solving Olympiad geometry problems. To achieve this, we …
Language ModelingLanguage ModellingMathSynthetic Data GenerationLearning with symmetric positive definite matrices via generalized Bures-Wasserstein geometry
Learning with symmetric positive definite (SPD) matrices has many applications in machine learning. Consequently, understanding the Riemannian geometry of SPD matrices has attracted much attention lately. A particular Ri…
Riemannian optimizationAchieving Olympiad-Level Geometry Large Language Model Agent via Complexity Boosting Reinforcement Learning
Large language model (LLM) agents exhibit strong mathematical problem-solving abilities and can even solve International Mathematical Olympiad (IMO) level problems with the assistance of formal proof systems. However, du…
Reinforcement LearningProposing and solving olympiad geometry with guided tree search
Mathematics olympiads are prestigious competitions, with problem proposing and solving highly honored. Building artificial intelligence that proposes and solves olympiads presents an unresolved challenge in automated the…
Bi-FlowGS: Bridging Generative View Completion and Gaussian Geometry through Bidirectional Flow Co-Refinement
Sparse-view 3D scene reconstruction with 3D Gaussian Splatting (3DGS) is inherently underconstrained. Plausible renderings can also coexist with erroneous Gaussian geometry, as errors in positions or depths may be concea…
Video Restoration