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Multidimensional Scaling (MDS) — variants.

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Answer

  • Given pairwise distance matrix D, find low-dim embedding preserving distances.
  • Classical MDS: closed form via eigendecomposition of doubly-centered D2D^{2} — equivalent to PCA on distance data.
  • Metric MDS: minimize stress = Σ (∣∣yi  −  yj∣∣  −  dij)2( \mid \mid y_{i}\; - \;y_{j} \mid \mid \; - \;d_{\mathrm{ij}})^{2}.
  • Non-metric MDS: preserves only rank order of distances.
  • Uses: psychometrics, marketing perception maps, small-n visualization.
Check yourself — multiple choice
  • Same as PCA
  • Given pairwise D, embed to preserve distances. Classical (=PCA  on  double−centered  D2)  /  metric( = \mathrm{PCA}\;\mathrm{on}\;\mathrm{double} - \mathrm{centered}\;D^{2})\; / \;\mathrm{metric} (stress) / non-metric (rank-preserving)
  • Random
  • Not real

MDS: preserve pairwise distances; classical / metric / non-metric variants.

#dimensionality-reduction

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