## Parallel Vector Field Embedding

*Binbin Lin, Xiaofei He, Chiyuan Zhang, Ming Ji*; 14(Oct):2945−2977, 2013.

### Abstract

We propose a novel local isometry based dimensionality reduction
method from the perspective of vector fields, which is called
parallel vector field embedding (PFE). We first give a
discussion on local isometry and global isometry to show the
intrinsic connection between parallel vector fields and
isometry. The problem of finding an isometry turns out to be
equivalent to finding orthonormal parallel vector fields on the
data manifold. Therefore, we first find orthonormal parallel
vector fields by solving a variational problem on the manifold.
Then each embedding function can be obtained by requiring its
gradient field to be as close to the corresponding parallel
vector field as possible. Theoretical results show that our
method can precisely recover the manifold if it is isometric to
a connected open subset of Euclidean space. Both synthetic and
real data examples demonstrate the effectiveness of our method
even if there is heavy noise and high curvature.

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