Embedding Riemannian manifolds by their heat kernel

Abstract

By embedding a class of closed Riemannian manifolds (satisfying some curvature assumptions and with diameter bounded from above) into the same Hilbert space, we interpret certain estimates on the heat kernel as giving a precompactness theorem on the class considered.

Critical sets of random smooth functions on compact manifolds

“…More precisely, they showed that the integral in the right-hand side of the above equality is the the total curvature of the immersion given by the evaluation map 5) where ev p (u) = u(p), ∀u ∈ U L . For our purposes the probabilistic description of the integrand ρ L (x) is more useful.…”

Hierarchical Matching of Non-rigid Shapes

“…Since diffusion distances are derived from the Laplace Beltrami operator, they are also intrinsic properties, and, according to [3,11,10], also fulfill the metric axioms.…”

Shape matching based on diffusion embedding and on mutual isometric consistency

“…The diffusion kernel is the fundamental solution to the diffusion equation and can be represented in terms of the eigenvalues and eigenfunctions of the Laplacian operator. This leads to an important intrinsic shape representation known as diffusion-kernel embedding [1]. The diffusion kernel also allows the definition of the diffusion distance which is an interesting alternative to the geodesic distance and which can be used in various ways such as: building invariant shape signatures, i.e., distance distributions [21,8], computing differential properties of meshes and of point clouds [19], or comparing shapes [5,23].…”