Theoretical analysis of LLE based on its weighting step

Yair Goldberg, Ya'acov Ritov

Research output: Contribution to journalArticlepeer-review


The local linear embedding algorithm (LLE) is a widely used nonlinear dimensionreducing algorithm. However, its large sample properties are still not well understood. In this article, we present new theoretical results for LLE based on the way that LLE computes its weight vectors. We show that LLE's weight vectors are computed from the high-dimensional neighborhoods and are thus highly sensitive to noise. We also demonstrate that in some cases LLE's output converges to a linear projection of the highdimensional input. We prove that for a version of LLE that uses the low-dimensional neighborhood representation (LDR-LLE), the weights are robust against noise. We also prove that for conformally embedded manifold, the preimage of the input points achieves a low value of the LDR-LLE objective function, and that close-by points in the input are mapped to close-by points in the output. Finally, we prove that asymptotically LDR-LLE preserves the order of the points of a one-dimensional manifold. The Matlab code and all datasets in the presented examples are available as online supplements.

Original languageEnglish
Pages (from-to)380-393
Number of pages14
JournalJournal of Computational and Graphical Statistics
Issue number2
StatePublished - Jun 2012
Externally publishedYes

Bibliographical note

Funding Information:
This research was supported in part by the Israeli Science Foundation grant 209/6. We are grateful to the anonymous reviewers of early versions of this article for their helpful suggestions. Helpful discussions with Alon Zakai and Jacob Goldberger are gratefully acknowledged.


  • Dimension reduction
  • Locally linear embedding
  • Manifold learning

ASJC Scopus subject areas

  • Statistics and Probability
  • Discrete Mathematics and Combinatorics
  • Statistics, Probability and Uncertainty


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