Many machine learning algorithms for clustering or dimensionality re- duction take as input a cloud of points in Euclidean space, and construct a graph with the input data points as vertices. This graph is then parti- tioned (clustering) or used to redefine metric information (dimensional- ity reduction). There has been much recent work on new methods for graph-based clustering and dimensionality reduction, but not much on constructing the graph itself. Graphs typically used include the fully- connected graph, a local fixed-grid graph (for image segmentation) or a nearest-neighbor graph. We suggest that the graph should adapt locally to the structure of the data.