202412112156
Status: #idea
Tags:
State: #nascient

Hyperplane

In a $p -$ dimensional space, a hyperplane is a flat affine subspace of dimension $p - 1$ .
By their very definition, they split a $p$ dimensional space into two parts as there's only one axis left. The Maximal Margin Classifier (Optimal Separating Hyperplaene) and its extension the Support Vector Machine (SVM) both make use of this fact.

In general, the equation for such a plane is:

β_{0} + β_{1} X_{1} + β_{2} X_{2} + \dots + β_{p} X_{p} = 0

You check that a point lies on the hyperplane by plugging the point in the equation and seeing if the equation is satisfied.

By that logic if we assume that

β_{0} + β_{1} X_{1} + β_{2} X_{2} + \dots + β_{p} X_{p} < 0

this tells us that $X$ lies below the hyperplane, on the other hand if we get:

β_{0} + β_{1} X_{1} + β_{2} X_{2} + \dots + β_{p} X_{p} > 0

the point lies atop it.

It is typically a good idea to check where the $0$ vector lies to have a better idea of the space.

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