A Global Geometric Framework for Nonlinear
Dimensionality Reduction

J. B. Tenenbaum, V. de Silva and J. C. Langford

Science 290 (5500): 2319-2323, 22 December 2000

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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).