A Global Geometric Framework for Nonlinear
J. B. Tenenbaum, V. de Silva and J. C. Langford
Science 290 (5500): 2319-2323, 22 December 2000
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 defined in the Euclidean metric. As shown in
Fig. 2C of the paper, 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).