C here is working as a tuning parameter that is chosen via cross-validation. Now comes the trade-off between bias and variance. So, the thing to be consider is that for C > 0 no more than C observation can wrong side of the hyperplane. Be careful that the value of slack variable can be in fraction as well, for example, let’s say C=0.6, then it might the possible that we have three points with slack variable = 0.2 or two points with slack variable = 0.3 or we have C=1.6 with one slack variable =0.6 (wrong side of margin) and another slack variable= 1 (wrong side of the plane). If C=0 it means no observation has violated the margin. Now consider the role of variable C, it is the sum of all the slack variables for all n observations. If slack variable is greater than 1 then “i th” observation is on the wrong side of the hyperplane. If the slack variable is equal to zero then it means that “i th” observation is on correct side of margin and if slack variable is > than 0 then it means “i th” observation is on wrong side of margin.
Meaning, if you move vectors, the classifier would change its margin, but if you move all other observations the margin wouldn’t change. Interestingly, this classifier simply depends upon those vectors and not on all other observations available in the training set.
Those touching points are called vectors because they are vectors in p-dimensional space.
They are defining the length of the margin, farther those points are, farther away those two lines are from the middle hyperplane. Those lines are called margins, and the observations touching those two lines are called vectors. The way maximal margin classifier looks like is that it has one plane that is cutting through the p-dimensional space and dividing it into two pieces, and then it has two lines, each on one and other side of that plane. In simple words, for each testing observation we put all the variables in the equation above and decide which side of the hyperplane that particular observation belongs to, based on the sign of f(x).
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