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CNA Seminar/Colloquium/Joint Pitt-CNA Colloquium
Facundo Memoli
Ohio State
Title: A spectral notion of distance between shapes

Abstract: I'll discuss the construction of a spectral notion of distance between shapes (closed Riemannian manifolds). This distance satisfies the properties of a metric on the class of isometric shapes, which means, in particular, that two shapes are at 0 distance if and only if theyare isometric.

The construction is similar to the Gromov-Wasserstein distance between two metric measure spaces, but rather than viewing the two shapes merely as mm-spaces, one defines the distance via the comparison of their heat kernels.

This permits relating this distance to previously proposed spectral invariants used for shape comparison, such as the spectrum of the Laplace-Beltrami operator and statistics of diffusion distances. In addition, the heat kernel provides a natural notion of scale, which is useful for multi-scale shape comparison. This distance admits a hierarchy of lower bounds which provide increasing discriminative power at the cost of an increase in computational complexity. As a result, it is possible to write several quantitative estimates for the distance between canonical shapes (such as spheres of different dimension) -- something which seems to be harder in the purely metric setting.

Date: Thursday, April 30, 2015
Time: 4:30 pm
Location: Wean Hall 7218
Submitted by:  Slepcev