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2.6. GENERATIVE DIMENSIONALITY REDUCTION 35
X
Y
Figure 2.3: The generative latent variable model. The observed data X is modeled
as generated from a low-dimensional latent variable Y through the generative
mapping f specified by parameters W.
been rolled up into a ball. The string is a one dimensional object embedded in a
three dimensional space. Locality is preserved through the generating mapping
f, i.e. points that are close on the string will remain close in the ball. However,
the reverse does not necessarily need to be true as neighboring points on the ball
W
might come from different “loops” when the ball was rolled together.
Further, even though assumed to exists, none of the spectral algorithms models
the smooth inverse of the generating mapping but rather learns embeddings of the
data points this is fine as long as focus is on the visualization of the data rather
then a model.
2.6 Generative Dimensionality Reduction
Generative approaches to dimensionality reductions aim to model the observed
data as a mapping from its intrinsic representation. The underlying representa-
tion is often referred to as the latent representation of the data and the models as
latent variable models for dimensionality reduction. The observed data Y have
been generated from the latent variables X through a mapping f parameterized