One of the most popular algorithms for nonlinear dimensionality reduction, called Isomap, does just this. It connects each data point in a high-dimensional space (a face, say) to all nearby points (very similar faces), computes the shortest distances between all pairs of points along the resulting network and finds the reduced coordinates that best approximate these distances. In contrast to PCA, faces’ coordinates in this space are often quite meaningful: one may represent which direction the face is facing (left profile, three quarters, head on, etc.); another how the face looks (very sad, a
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