We address the problem of setting the kernel bandwidth, epps, used by Manifold Learning algorithms to construct the graph Laplacian. Exploiting the connection between manifold geometry, represented by the Riemannian metric, and the Laplace-Beltrami operator, we set epps by optimizing the Laplacian's ability to preserve the geometry of the data. Experiments show that this principled approach is effective and robust

Many manifold learning algorithms aim to create embeddings with low or no distortion (i.e. isometric). If the data has intrinsic dimension d, it is often impossible to obtain an isometric embedding in d dimensions, but possible in s > d dimensions. Yet, most geometry preserving algorithms cannot do the latter. This paper proposes an embedding algorithm that overcomes this problem. The algorithm directly computes, for any data embedding Y, a distortion loss(Y), and iteratively updates Y in order to decrease it. The distortion measure we propose is based on the push-forward Riemannian metric associated with the coordinates Y. The experiments confirm the superiority of our algorithm in obtaining low distortion embeddings.

This paper considers the problem of embedding directed graphs in Euclidean space while retaining directional information. We model the observed graph as a sample from a manifold endowed with a vector field, and we design an algo- rithm that separates and recovers the features of this process: the geometry of the manifold, the data density and the vector field. The algorithm is motivated by our analysis of Laplacian-type operators and their continuous limit as generators of diffusions on a manifold. We illustrate the recovery algorithm on both artificially constructed and real data.