O UT - OF -S AMPLE E XTENSION AND R ECONSTRUCTION ON M ANIFOLDS Bhuwan Dhingra Final Year (Dual Degree) Dept of Electrical Engg.

I NTRODUCTION 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

I NTRODUCTION n >> m

N ON -L INEAR D IMENSIONALITY R EDUCTION 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

P ROBLEM S TATEMENT x* y* g f

O UTLINE

The tangent plane is estimated from the k- nearest neighbors of p using PCA

O UT - OF -S AMPLE E XTENSION A linear transformation A e is learnt s.t Y = A e Z Embedding for new point y* = A e z* AeAe z*y*

O UT - OF -S AMPLE R ECONSTRUCTION A linear transformation A r is learnt s.t Z = A r Y Projection of reconstruction on tangent plane z* = A r y* z*y* ArAr

P RINCIPAL C OMPONENTS A NALYSIS Covariance matrix of neighborhood: Let be the eigenvector and eigenvalue matrixes of M k Then Denotethen the projection of a point x onto the tangent plane is given by:

L INEAR T RANSFORMATION

F INAL E STIMATES

E RROR A NALYSIS

S AMPLING D ENSITY

N EIGHBORHOOD P ARAMETERIZATION

R ECONSTRUCTION E RROR But A r A e = I, hence

R ECONSTRUCTION E RROR

S MOOTHNESS OF M ANIFOLD

R ESULTS - E XTENSION Out of sample extension on the Swiss-Roll dataset Neighborhood size = 10

R ESULTS - E XTENSION Out of sample extension on the Japanese flag dataset Neighborhood size = 10

R ESULTS - R ECONSTRUCTION Reconstructions of ISOMAP faces dataset (698 images) n = 4096, m = 3 Neighborhood size = 8

R ECONSTRUCTION ERROR V N UMBER OF P OINTS ON M ANIFOLD ISOMAP Faces dataset Number of cross validation sets = 5 Neighborhood size = [6, 7, 8, 9]