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2.4. SPECTRAL DIMENSIONALITY REDUCTION 23
with the dimensionality of,
rank(X) = rank(XX T ) = rank(G) = rank(D(X)) = d.
In practice, for dimensionality reduction, we want to find a low dimensional repre-
sentation of a set of data points i.e. vectorial data. In this case the Gram matrix G
can be constructed directly from the data and a rank d dimensional approximation
can be sought making the conversion step from distance matrix to gram matrix
uneccesary.
Principal Component Analysis (PCA) is a dimensionality reduction technique
for embedding vectorial data in a dimensionally reduced representation. Given
centered vectorial data Y the covariance matrix S = Y T Y has elements on the
diagonal representing the variance along each dimension of the data while the off-
diagonal elements measures the linear redundancies between dimensions. The
objective of PCA is to find a projection v of the data Y such that the variance
along each dimension is maximized,
Objective: argmax v var(Yv) (2.19)
subject to: v T v = 1. (2.20)
This implies finding a projection of the data into a representation resulting in a