# Mathematical formulation of Support Vector Machines?

I'm trying to learn maths behind SVM (hard margin) but due to different forms of mathematical formulations I'm bit confused.

• Assume we have two sets of points $$\text{(i.e. positives, negatives)}$$ one on each side of hyperplane $$\pi$$.

• So the equation of the margin maximizing plane $$\pi$$ can be written as, $$\pi:\;W^TX+b = 0$$

• If $$y\in$$ $$(1,-1)$$ then,

$$\pi^+:\; W^TX + b=+1$$ $$\pi^-:\; W^TX + b=-1$$

• Here $$\pi^+$$ and $$\pi^-$$ are parallel to plane $$\pi$$ and they are also parallel to each other. Now the objective would be to find a hyperplane $$\pi$$ which maximizes the distance between $$\pi^+$$ and $$\pi^-$$.

Here $$\pi^+$$ and $$\pi^-$$ are the hyperplanes passing through positive and negative support vectors respectively

• According to wikipedia about SVM I've found that distance/margin between $$\pi^+$$ and $$\pi^-$$ can be written as, $$\hookrightarrow\frac{2}{||w||}$$

• Now if I put together everything this is the constraint optimization problem we want to solve, $$\text{find}\;w_*,b_* = \underbrace{argmax}_{w,b}\frac{2}{||w||} \rightarrow\text{margin}$$ $$\hookrightarrow \text{s.t}\;\;y_i(w^Tx\;+\;b)\;\ge 1\;\;\;\forall\;x_i$$

Before proceeding to my doubts please do confirm if my understanding above is correct? If you find any mistakes please do correct me.

• How to derive margin between $$\pi^+$$ and $$\pi^-$$ to be $$\frac{2}{||w||}?$$ I did find a similar question asked here but I couldn't understand the formulations used there? If possible can anyone explain it in the formulation I used above?
• How can $$y_i(w^Tx+b)\ge1\;\;\forall\;x_i$$?

deriving the margin to be $$\frac{2}{|w|}$$

we know that $$w \cdot x +b = 1$$

If we move from point z in $$w \cdot x +b = 1$$ to the $$w \cdot x +b = 0$$ we land in a point $$\lambda$$. This line that we have passed or this margin between the two lines $$w \cdot x +b = 1$$ and $$w \cdot x +b = 0$$ is the margin between them which we call $$\gamma$$

For calculating the margin, we know that we have moved from z, in opposite direction of w to point $$\lambda$$. Hence this margin $$\gamma$$ would be equal to $$z - margin \cdot \frac{w}{|w|} = z - \gamma \cdot \frac{w}{|w|} =$$ (we have moved in the opposite direction of w, we just want the direction so we normalize w to be a unit vector $$\frac{w}{|w|}$$)

Since this $$\lambda$$ point lies in the decision boundary we know that it should suit in line $$w \cdot x + b = 0$$ Hence we set is in this line in place of x:

$$w \cdot x + b = 0$$ $$w \cdot (z - \gamma \cdot \frac{w}{|w|}) + b = 0$$ $$w \cdot z + b - w \cdot \gamma \cdot \frac{w}{|w|}) = 0$$ $$w \cdot z + b = w \cdot \gamma \cdot \frac{w}{|w|}$$ we know that $$w \cdot z +b = 1$$ (z is the point on $$w \cdot x +b = 1)$$ $$1 = w \cdot \gamma \cdot \frac{w}{|w|}$$ $$\gamma= \frac{1}{w} \cdot \frac{|w|}{w}$$ we also know that $$w \cdot w = |w|^2$$, hence: $$\gamma= \frac{1}{|w|}$$ Why is in your formula 2 instead of 1? because I have calculated the margin between the middle line and the upper, not the whole margin.

• How can $$y_i(w^Tx+b)\ge1\;\;\forall\;x_i$$?

We want to classify the points in the +1 part as +1 and the points in the -1 part as -1, since $$(w^Tx_i+b)$$ is the predicted value and $$y_i$$ is the actual value for each point, if it is classified correctly, then the predicted and actual values should be same so their production $$y_i(w^Tx+b)$$ should be positive (the term >= 0 is substituded by >= 1 because it is a stronger condition)

The transpose is in order to be able to calculate the dot product. I just wanted to show the logic of dot product hence, didn't write transpose

For calculating the total distance between lines $$w \cdot x + b = -1$$ and $$w \cdot x + b = 1$$:

Either you can multiply the calculated margin by 2 Or if you want to directly find it, you can consider a point $$\alpha$$ in line $$w \cdot x + b = -1$$. then we know that the distance between these two lines is twice the value of $$\gamma$$, hence if we want to move from the point z to $$\alpha$$, the total margin (passed length) would be: $$z - 2 \cdot \gamma \cdot \frac{w}{|w|}$$ then we can calculate the margin from here.

derived from ML course of UCSD by Prof. Sanjoy Dasgupta

• "we know that we have moved from z, in direction of w to point λ" Here z lies in ($w^t.x+b=+1$.) we also know that normal(w) to the plane $\pi$ is in direction of positive points right? So aren't we moving from z to point $\lambda$ in opposite direction of w? – user_6396 Jul 12 '19 at 23:50
• How can I calculate whole margin so that formula becomes $\frac{2}{||w||}$? It will give me an intuition. – user_6396 Jul 13 '19 at 0:15
• I added it to the bottom of my post. – Fatemeh Asgarinejad Jul 13 '19 at 0:58
• I really appreciate all the help. Thanks :) – user_6396 Jul 13 '19 at 1:03
• I'm happy that I could help. :) – Fatemeh Asgarinejad Jul 13 '19 at 1:03