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

2.5. NON-LINEAR 33
forced to be centered,
i xi = 0. The optimal embedding ˆ X can be found
through an eigenvalue problem.
Laplacian Eigenmaps
The proximity graph is also the starting point for Laplacian eigenmaps [5].
Each node in the graph is connected to its neighbors by a vertex with an
edge weight representing the locality of the points. Several different mea-
sures of locality can be used. In the original paper either a heat kernel,
wij = e − ||y i −y j ||2 2
t , or constant wij = 1 was applied. Once the graph have
been constructed the objective is to find an embedding X of the data such
that points that are connected in the graph stay as close together as possible.
For the first dimension,
ˆX = argmin
(xi − xj)Wij = y T Ly, (2.37)
i,j
where L is referred to as the Laplacian defined as L = D − W and D is
a diagonal matrix such that Dii =
j Wji. The objective Eq. 2.37 has a
trivial solution zero dimensional solution representing the embedding using
a single point. To remove this solution the solution is forced to be orthogonal
to the constant vector 1, y T D1 = 0. Further, to shrinking the embedding a
constraint on the scale y T Dy = 1 is appended to the objective. The diagonal
matrix D provides a scaling of each point with respect to its locality to other
points in the data. For a multi-dimensional embedding of the data this leads