Every Stable Invariant of Finite Metric Spaces Produces False Positives
Explore the limitations of stable invariants in computational topology and geometry through this 56-minute lecture by Nicolò Zava. Delve into the Gromov-Hausdorff distance framework for shape recognition and comparison, and understand why its direct computation for finite metric spaces is NP-hard. Learn about the approach of using invariants to approximate the Gromov-Hausdorff distance and the requirement for stability in these invariants. Discover the groundbreaking conclusion that false positives are unavoidable when stable invariants take values in a Hilbert space, derived from the proof that the space of isometry classes of finite metric spaces with the Gromov-Hausdorff distance cannot be coarsely embedded into any Hilbert space. Gain insights into the implications of this finding for the field of computational topology and its applications in shape analysis.