Curvature Estimation for Ricci Flow Embedding

This paper makes two contributions to the problem of correcting non-Euclidean dissimilarities. Data of this sort arrise when there are negative Eigen values of the dissimilarity matrix, and can therefore not be embedded into a real-valued Euclidean space. Our first contribution is to show how the non-Euclidean artifacts can be rectified. This is achieved by applying Ricci flow to the embedding of the manifold on which the data reside, and performing reprojection of the geodesic distances from tangent spaces centred on local patches of the manifold. Our second contribution is to show how the curvature of the local patches needed in the reprojection, can be estimated from the local point density on the patches. We experiment with the method on the well known Chicken pieces dataset [4].