# Geometric interpretation of support vector values in primal space

The Linear Support Vector Machine classification ($y_{k} = -1\ \mathrm{or}\ +1$) with misclassification tolerance loss function in primal weight space looks like this:

$$\min\limits_{w,b,\xi} J_{P}(w,\xi) = \frac{1}{2}w^{T}w + c\sum\limits_{k=1}^{N}\xi_{k}$$

Subject to conditions: $$\forall_{k\in1...N} \ \ \xi_{k} \geq 0$$ $$\forall_{k\in1...N} \ \ y_{k}(w^{T}x_{k}+b) \geq 1 - \xi_{k}$$

In dual space it becomes:

$$\max\limits_{\alpha} J_{D}(\alpha) = -\frac{1}{2}\sum\limits_{k,l=1}^{N} y_{k}y_{l}x_{k}^{T}x_{l}\alpha_{k}\alpha_{l}+\sum\limits_{k=1}^{N}\alpha_{k}$$

Subject to conditions: $$\sum\limits_{k=1}^{N}\alpha_{k}y_{k} = 0$$ $$\forall_{k=1...N}\ \ 0 \leq \alpha_{k} \leq c$$

My geometric intepretation of these values:

So I can say that (correct me if I'm wrong):

$w^{T}x+b = 0$ is the decision boundary line.

$w^{T}x+b= -1, 1$ are the margins for respective classes.

$\xi_{k}$ (slack variables) are distances from the margin of correct classification for $k$ data point.

My question is:

Are there geometric interpretations of $c$ and $\alpha$ values which can be visualised on the above pictorial interpretation as well? If so, what are they?

Your picture is almost correct. Qualitatively, it is correct, however, your drawing assumes that $\|w\| = 1$ which is not true in general. If you substituted the distance of $1$ with $\frac{1}{\|w\|}$ the drawing would be correct. Regarding the geometry of $c$ and $\alpha$..
$c$ just affects the mixture of regularization with margin violation. Geometrically, therefore it affects the distance between the two margins (i.e. smaller $c$ increases the distance between the margins because it makes $\|w\|$ smaller).
Because $\alpha$ exists in the dual problem, I am not aware of an interpretation in the primal problem other than the the KKT conditions:
$$(\alpha_k = 0) \implies y_{k}(w^{T}x_{k}+b) \gt 1$$ $$(0 \lt \alpha_k \lt c) \implies y_{k}(w^{T}x_{k}+b) = 1$$ $$(\alpha_k = c) \implies y_{k}(w^{T}x_{k}+b) \lt 1$$