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

2.5. NON-LINEAR 31
the graph is maximized under the constraint that the distance between near-
est neighbors stay the same. In effect the MVU objective will try to unravel
the data by stretching the manifold as much as possible without causing it to
tear. Rather then formulating the objective in terms of a distance matrix as
in Isomap the MVU objective is expressed in terms of the Gram matrix which
we know are interchangeable representations Eq. 2.17. MVU tries to find a
feature space represented by a kernel matrix K. This leads to the following
objective,
ˆK = argmax tr(K) (2.34)
subject to: K 0
Kij = 0
ij
Kii − Kjj − Kij − Kji = Gii − Gjj − Gij − Gji, i ∈ N(j),
where G is the Gram matrix for the original representation and N(i) is the
index set of points that are connected to i in the proximity graph. The first
constraint forces the ˆ K to represent a geometrically interpretable feature
space, while the second constraint forces the data to be centered. The fi-
nal constraint ensures that the distance between points that are connected in
the proximity graph is conserved. The optimization is an instance of Semi-
Definite Programming [11] and can be solved using efficient algorithms. Once
a valid kernel matrix K has been found the resulting embedding can be found
by applying MDS to ˆ K.