A Global Geometric Framework for Nonlinear Dimensionality Reduction Joshua B. Tenenbaum, Vin de Silva, and John C. Langford

Supplementary Material

Supplemental Figure 1. Isomap (K = 6) applied to N = 2000 images (64 pixels by 64 pixels) of a hand in different configurations. The images were generated by making a series of opening and closing movements of the hand at different wrist orientations, designed to give rise to a two-dimensional manifold. The images were treated as 4096-dimensional vectors, with input-space distances d_X(i,j) defined in the Euclidean metric. As shown in Fig. 2C, Isomap correctly detects two clearly significant dimensions, plus several weak dimensions of noise; PCA and MDS do not detect the correct dimensionality and suggest a much higher level of noise. The recovered coordinate axes map approximately onto the distinct underlying degrees of freedom: wrist rotation (x axis) and finger extension (y axis).

Medium version | Full size version