Separate a normal’s direction from its length

THE IDEA

Read the example.

The vector w is normal to a linear decision boundary. Its unit version gives a direction independent of scale. Dividing a point’s functional score by ‖w‖ converts that score into signed perpendicular distance.

Distance = |wᵀx + b| / ‖w‖₂

Use it well

Multiplying both w and b by a positive constant preserves the boundary but changes the raw score. A raw score is therefore not automatically a distance or probability.