When using the eigenmap based on ϒϒt a formal test for the number of significant dimensions can be based on the theoretical results concerning the Tracy-Widom distribution of eigenvalues of a covariance matrix [Johnstone, 2001; Patterson et al., 2006]. Tracy-Widom theory does not extend to the eigenvalues of the graph Laplacian. Instead we introduce a different approach, known as the eigengap heuristic, based on the difference in magnitude between successive eigenvalues.
