Manifold learning: The price of normalization

Yair Goldberg, Alon Zakai, Dan Kushnir, Ya'acov Ritov

Research output: Contribution to journalArticlepeer-review


We analyze the performance of a class of manifold-learning algorithms that find their output by minimizing a quadratic form under some normalization constraints. This class consists of Locally Linear Embedding (LLE), Laplacian Eigenmap, Local Tangent Space Alignment (LTSA), Hessian Eigenmaps (HLLE), and Diffusion maps. We present and prove conditions on the manifold that are necessary for the success of the algorithms. Both the finite sample case and the limit case are analyzed. We show that there are simple manifolds in which the necessary conditions are violated, and hence the algorithms cannot recover the underlying manifolds. Finally, we present numerical results that demonstrate our claims.

Original languageEnglish
Pages (from-to)1909-1939
Number of pages31
JournalJournal of Machine Learning Research
StatePublished - Aug 2008
Externally publishedYes


  • Diffusion maps
  • Dimensionality reduction
  • Hessian eigenmap
  • Laplacian eigenmap
  • Local tangent space alignment
  • Locally linear embedding
  • Manifold learning

ASJC Scopus subject areas

  • Software
  • Control and Systems Engineering
  • Statistics and Probability
  • Artificial Intelligence


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