# In Support-Vector-Machine, why is the hyperplane given by $(p-1)$ dimensions?

In SVM, each feature vector is viewed as a data point in a $$p$$-dimensional plane and different labels are separated by a $$(p-1)$$ dimensional hyperplane.

Please see the wikipedia entry: https://en.wikipedia.org/wiki/Support-vector_machine#Motivation

My question is:

• Why does the hyperplane have $$(p-1)$$ dimensions and not just $$p$$?
• If you connect 2 points in $n$-dimensional space by a line, the line is a 1-dimensional object. That line will traverse a region that has a limited extent in 1 dimension (the dimension of the line). Now we can take a point on that line between the two original points. From that "in between point" lets extend out in all other dimensions as far as we can (to $\infty$). This is now a $n-1$ dimensional space (aka hyperplane). The original 2 points are "separated" by this hyperplane. If the data is somewhat concentrated near these two original points, then it to can be separated by this hyperplane. – jdods Feb 22 at 21:28
• @Carser, oh god, I just had a brain-fart.. I was convinced that a line was a 2 dimensional object for some reason. God damn it, thanks! – iamatrain Feb 22 at 21:33

In the simplest case you have datapoints from a $$1$$ dimensional set, which you can represent as points on a line (think like the number line), you could separate these points with one point. For concretenss sake you can imagine having your dataset describing weights of mice ranging from $$85$$ grams to $$245$$ grams and you say that all mice which has weight above $$100$$ grams are categorised as "overweight" so you can set a point down at $$100$$ and say that everything above is overweight and below is not overweight. Think what we have just done, we separated (classified) a $$1$$ dimensional dataset with a $$0$$ dimensional object, a point. It wouldn´t make sense to separate a line with a $$1$$ dimensional object.
When you have a $$2$$ dimensional data e.g. a plane of points here you would´t be able to separate anything with a point i.e. a $$0$$ dimensional object but a line or curve would do perfectly which is in turn a $$1$$ dimensional object. Once again a plane would not work as a separator since from the datasets "point of view" nothin else exists than this plane.
A linear classifier of dimension $$p$$ in a $$p$$-dimensional space would contain all data points. If it were of dimension smaller than $$p-1$$ then it wouldn't separate the space into two regions. It wouldn't classify. A $$(p-1)$$-dimensional hyperplane separates the space of possible data points into two disjoint half spaces.