Explaining Decision Boundary of a Support Vector Machine 🌱

Last updated on August 10, 2020

Consider the equation below which has relation to the primal problem of SVMs:

h (w, b) = max_{α_{i} \geq 0 \forall i} \sum_{i = 1}^{N} α_{i} (1 - y_{i} (⟨ w, ϕ (x_{i}) ⟩ + b))

Specifically let’s examine $y_{i} (⟨ w, ϕ (x_{i}) ⟩ + b)$ :

⟨ w, ϕ (x_{i}) ⟩ + b = 0 represents the decision boundary line

For all ϕ (x_{i}) lying on the same side of the line as the vector w, ⟨ w, ϕ (x_{i}) ⟩ + b > 0

For all ϕ (x_{i}) lying on the side of the line opposite the vector w, ⟨ w, ϕ (x_{i}) ⟩ + b < 0

b shifts the location of the line and w changes the slope

Correct classification of a point:

y_{i} (⟨ w, ϕ (x_{i}) ⟩ + b) > 0 for a point where y_{i} is the same sign as (⟨ w, ϕ (x_{i}) ⟩ + b)

Incorrect classification:

y_{i} (⟨ w, ϕ (x_{i}) ⟩ + b) < 0 for a point where y_{i} is a different sign than (⟨ w, ϕ (x_{i}) ⟩ + b)

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