A Global Geometric Framework for Nonlinear Dimensionality Reduction

A Global Geometric Framework for Nonlinear Dimensionality Reduction
Joshua B. Tenenbaum,1* Vin de Silva,2 John C. Langford3
Science 2000
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Below is  canonical problems in dimensionality reduction from the domain of visual perception

The paper proposed a method called Isomap, it builds on classical MDS but seeks to preserve the intrinsic geometry of the data, as captured in the geodesic manifold distances between all pairs of data pointsthe. Algorithm has three steps as below:

So we could simply see Isomap as an approach that combines the major algorithmic features of PCA

and MDS, but calculate the geodesic distance in advance. The figure below shows the physical meaning of geodesic distance and the sampling effect.

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Sumary and Comment:

Compared to MDS: ISOMAP has the ability to discover the underlying structure (latent variables) which is nonlinear embedded in the feature space. It is a global method, which preserves all pairs of
distances.