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OUT-OF-SAMPLE EXTENSION AND RECONSTRUCTION ON MANIFOLDSBhuwan DhingraFinal Year (Dual Degree)Dept of Electrical Engg.
INTRODUCTION
An m-dimensional manifold is a topological space which is locally homeomorphic to the m-dimensional Euclidean space
In this work we consider manifolds which are: Differentiable Embedded in a Euclidean space Generated from a set of m latent variables via a
smooth function f
NON-LINEAR DIMENSIONALITY REDUCTION
In practice we only have a sampling on the manifold
Y is estimated using a Non-Linear Dimensionality Reduction (NLDR) method
Examples of NLDR methods –ISOMAP, LLE, KPCA etc.
However most non-linear methods only provide the embedding Y and not the mappings f and g
OUTLINE
p is the nearest neighbor of x* Only the points in are used for extension and
reconstruction
OUT-OF-SAMPLE EXTENSION
A linear transformation Ae is learnt s.t Y = AeZ
Embedding for new point y* = Aez*
�̂�𝑝∈𝑍 𝑦 𝑝∈𝑌Ae
z* y*
OUT-OF-SAMPLE RECONSTRUCTION
A linear transformation Ar is learnt s.t Z = ArY
Projection of reconstruction on tangent plane z* = Ary*
�̂�𝑝∈𝑍 𝑦 𝑝∈𝑌
z* y*Ar
PRINCIPAL COMPONENTS ANALYSIS
Covariance matrix of neighborhood:
Let be the eigenvector and eigenvalue matrixes of Mk
Then
Denote then the projection of a point x onto the tangent plane is given by:
LINEAR TRANSFORMATION
Y and Z are both centered around and Then Ae =BeRe where Be and Re are scale and
rotation matrices respectively If is the singular value decomposition
of ZTY, then
ERROR ANALYSIS
We don’t know the true form of f or g to compare our estimates against
Reconstruction Error: For a new point x* its reconstruction is computed as , and the reconstruction error is
SAMPLING DENSITY
To show: As the sampling density of points on the manifold increases, reconstruction error of a new point goes to 0
In a k-NN framework, the sampling density can increase in two ways: k remains fixed and the sampling width
decereases remains fixed and
We consider the second case
RECONSTRUCTION ERROR
Tyagi, Vural and Frossard (2012) derive conditions on k s.t the angle between and is bounded
They show that as
Equivalently, where Rm is an aribitrary m-dimensional rotation matrix
and
SMOOTHNESS OF MANIFOLD
If the manifold is smooth then all will be smooth
Taylor series of :
As because x* will move closer to p
RESULTS - RECONSTRUCTION
Reconstructions of ISOMAP faces dataset (698 images)
n = 4096, m = 3 Neighborhood size = 8