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Learn optimal reach estimation techniques for embedded submanifolds in this 51-minute conference talk from Harvard CMSA's Conference on Geometry and Statistics. Explore the concept of reach as introduced in Federer's foundational work on curvature measures, understanding it as a scale parameter that determines when a submanifold is sufficiently flat for applying traditional Euclidean statistical techniques with controlled approximation error. Discover multiple approaches for estimating reach from sample data on submanifolds, including methods that achieve optimality from a minimax estimation theory perspective. Examine intermediate estimation problems that arise in the process, including curvature estimation, weak feature size estimation, and distance estimation, while analyzing various statistical phenomena such as different convergence rates and inconsistency issues. Gain insights into the current state of research in geometric statistics through this comprehensive overview of theoretical developments and practical estimation challenges in submanifold analysis.
