Figure 1.

Illustrative sketch in two dimensions. While the data space is two dimensional the data points (circles) cluster around a one-dimensional manifold. While the Euclidian distance between points may be a good way to quantify close points it fails on larger scales. For example, point A seems to be closer to B than to C in terms of Euclidian distance, but quantifying distance in terms of distance along the manifold (which is likely the actual distance A would have to travel to ‘become like B’) reveals that A is closer to C than to B.