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2.4. SPECTRAL DIMENSIONALITY REDUCTION 21
matrices,
Aij =
N
k=1
VikΛkk
V T
kj =
N
(vk)iλk(vk)j =
k=1
N
λkvkv T
k ij .(2.13)
As V specifies an orthonormal basis the relative magnitude of each eigenvalue
corresponds to the the amount of A that is explained by the corresponding eigen-
vector. Therefore, by ordering the eigenvalues in decreasing order we can refer
to,
A→i =
i
k=1
as the best rank i approximation of matrix A.
k=1
λkvkv T k , (2.14)
λi ≥ λj, i ≤ j
2.4 Spectral Dimensionality Reduction
Spectral dimensionality reduction is based on the assumption that the generating
mapping f is invertible. This means that the relationship between the observed
representation Y and the intrinsic representation X takes the form of a bijection.
This implies that the intrinsic structure of the data is fully preserved in the ob-
served representation.
Classic Multi Dimensional Scaling (MDS) [17, 40] is a method for represent-
ing a metric dissimilarity measure as a geometrical configuration. Given dissim-
ilarity measure δij between i and j the aim is to find a geometrical configuration
of points X = [x1, . . .,xN] such that the Euclidean distance dij = ||xi − xj||2