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Locally Linear Embedding (LLE) — how does it work?

hard

Answer

  • (1) Find k nearest neighbors for each point.
  • (2) Reconstruct each point as linear combination of neighbors: minimize ||xi    Σx_{i}\; - \;{\Sigma} wijw_{\mathrm{ij}} xjx_{j}||² with Σ wij  =  1w_{\mathrm{ij}}\; = \;1.
  • (3) Find embedding yiy_{i} minimizing ||yi    Σy_{i}\; - \;{\Sigma} wijw_{\mathrm{ij}} yjy_{j}||² preserving the same weights.
  • Captures non-linear structure with a linear reconstruction locally.
  • Sensitive to k.
  • Modern: superseded by UMAP but still cited in manifold-learning theory.
Check yourself — multiple choice
  • Same as PCA
  • Reconstruct each point from neighbors with weights w; embed preserving same weights → non-linear DR with local linearity
  • Random
  • Not real

LLE: local linear reconstruction weights preserved in embedding.

#dimensionality-reduction

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