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What are hyperplanes?

What are hyperplanes? Here are some definitions.

Noun
  1. plural of hyperplane
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Examples
In their basic form, SVMs attempt to perform classification by constructing hyperplanes in a multidimensional space that separates the cases of different class labels.
In Section 4 we show that any PMD in which a coplane is not the intersection of two hyperplanes is an extension of a projective plane.
The heart of this paper is the following correspondence between Shi and Ish hyperplanes.
Its goal is to find hyperplanes in the attribute's space in order to maximize the margin between instances that belong to distinct classes.
It then employs hyperplanes to separate positive data from the negative ones.

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