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.

Nicolò Zava (3/17/23): Every stable invariant of finite metric spaces produces false positives