D_geo(i,j) = shortest path in k-NN graph
Isomap (Tenenbaum, de Silva, Langford 2000): approximate geodesic distances via shortest paths in k-NN graph, then apply MDS.
PCA (linear) fails on curved manifolds — it can't "unroll" the Swiss roll. Isomap preserves intrinsic geometry.
Top-left: 3D data (colored by latent coord).
Bottom-left: k-NN graph.
Top-right: PCA projection.
Bottom-right: Isomap embedding.