Figure 6.

Results of other variable-ID analysis methods on topologically complex artificial data. We consider sets of points drawn from mixtures of multivariate Gaussians embedded in curved nonlinear manifolds, as detailed in Fig. 2, and compared the results of Hidalgo (a), SMCE (b), LID (c). In panel (a) we plot again, for ease of comparison, the results of Fig. 2b. In panel (b), we show the local ID estimates given by LID¹⁰, considering $k=50$ nearest neighbors for local ID estimation and smoothing. The method assigns an integer dimension to each point: in the plot, the dimension of each point is represented by its color, according to the color bar shown. The method correctly retrieves the manifolds of dimension 5 and 9, even though the points in the manifold with dimension 5 are assigned a local ID that oscillates between 4 and 5, and points in the manifold with dimension 9 are assigned a local ID that oscillates between 7 and 8. Also, the manifold of dimension 2 is well identified. The manifolds of dimension 1 and 4, however, cannot be correctly discriminated. Points of the manifold of dimension 4 are assigned ID estimates of 1 or 3, without a clear separation of the manifolds. The NMI between the ground truth and the integer local ID is 0.77. In panel (c) we show the assignment of points as given by SMCE¹⁹. The manifolds of dimension 5 and 9 are merged. The manifold of dimension 2 is identified, but it is also contaminated by points from the manifold of dimension 4. SMCE cannot correctly retrieve the manifolds of dimension 1 and 4. The NMI between the ground truth and the assignment is 0.61. ID estimates obtained according to the prescription given in¹⁹ are largely incorrect.