The key quantity of interest here is the Riemannian metric, which characterizes the Riemannian geometry and defines straight lines and derivatives on the manifold.

Neural Information Processing Systems http://nips.cc/

Data from naturalistic systems are often modeled asgenerated from an underlying low dimensional manifold embedded inahighdimensional measurement space.

Isomap is a nonlinear dimensionality reduction method and finds low-dimensional embedding of high-dimensional data by preserving the pairwise geodesic distances between data pointsinmanifold. In2-dimensional embedding manifoldM,thegeodesic polygonal curvePi,j canbeprojected on the straight line connected it two endpoints. Every line segment ofPi,j has a corresponding line segmentinthethestraightline. The hyper-parameters searched over include the dimension of node representation as well as hyper-parameters specific to each model.

Geodesics play a critical role in many geometry processing applications.