Shared Gaussian Process Latent Variables Models - Oxford Brookes ...

2.5. NON-LINEAR 25
S. However, we also need to enforce orthogonality of the new basis,
(Y T vi) T (Y T vi) = v T i YYT vi = λi
1
√ (Y
λi
T vi) T (Y T vi) 1
√
λi
(2.29)
= v T i YYT vi = 1. (2.30)
This results in the eigen-basis of the covariance matrix i.e. v PCA
i
gives the following embedding,
x PCA
i
1
= YY T vi √
λi
=
= YT vi 1
N which
√ λi
N − 1 vi = 1
N − 1 xMDS
i , (2.31)
meaning MDS and PCA results in the same solution up to scale.
MDS and PCA assumes the generating mapping f to be linear and therefore
imply that the intrinsic representation of the data can be found by a change of basis
transform. This restricts these algorithms to only being applicable in scenarios
where the generating mapping is linear.
2.5 Non-Linear
Several algorithms have been suggested to model in the scenario where, rather
than assuming the generating mapping f to be linear, this is relaxed to only
assume that it is smooth. MDS finds a geometrical configuration respecting
a specific dissimilarity measure. Measuring the dissimilarity in terms of the
distance along the manifold between each point it would be possible to use
MDS even in scenarios where the generating mapping is non-linear. How-
ever, to acquire the distance along the manifold requires the manifold to be