Why does the linear SVM give a lot of support vectors?

Question

I simulate a simple linear setup:

n = 1000
X = runif(n)
Y = runif(n)

ind = X + 2*Y < 1
ind[ind == TRUE] = runif(sum(ind)) < 1
plot(X,Y,col = ind + 1)

Which gives

The svm() functcion from e1071 performs very well but it gives me a lot of vectors.

Call:
svm(formula = ind ~ X + Y, type = "C-classification", kernel = "linear")

Parameters:
   SVM-Type:  C-classification 
 SVM-Kernel:  linear 
       cost:  1 

Number of Support Vectors:  99

Can you please tell me what and how should I tune to get one vector (or just a few)?

Answer 1

Any linear SVM can be summarized as a single vector in input space:

$$f(\mathbf{z}) = \sum_{i \in SV} y_i \alpha_i \mathbf{x}_i^T\mathbf{z} +b,$$

can be rewritten as:

$$f(\mathbf{z}) = \mathbf{w}^T\mathbf{z} + b,$$

with

$$w[k] = \sum_{i\in SV} y_i \alpha_i x_i[k], \quad k=1..d$$

The amount of support vectors that actually form the model is not that relevant for a linear SVM, except for prediction speed (the above comment applies).

The problem here is that e1071 apparently uses LIBSVM instead of LIBLINEAR for linear SVM's. LIBSVM doesn't turn a linear model into a single vector in input space.