Editing Nonlinear dimensionality reduction (section)

=== Local tangent space alignment ===
{{Main|Local tangent space alignment}}
[[local tangent space alignment|LTSA]]<ref>{{Cite journal |last=Zhang |first=Zhenyue |author2=Hongyuan Zha  |title=Principal Manifolds and Nonlinear Dimension Reduction via Local Tangent Space Alignment |journal=SIAM Journal on Scientific Computing |volume=26 |issue=1 |year=2005 |pages=313–338 |doi=10.1137/s1064827502419154|citeseerx=10.1.1.211.9957 }}</ref> is based on the intuition that when a manifold is correctly unfolded, all of the tangent hyperplanes to the manifold will become aligned.  It begins by computing the ''k''-nearest neighbors of every point.  It computes the tangent space at every point by computing the ''d''-first principal components in each local neighborhood.  It then optimizes to find an embedding that aligns the tangent spaces.