A decision boundary
A threshold unit has w=(2,-1)^\top and bias w_0=-1. Describe the boundary, state which side gives output 1, and classify the points (1,0), (0,0), (1,2). What is the distance from the origin to the boundary?
Worked answer (click to show)
The boundary is 2x_1-x_2-1=0, that is x_2=2x_1-1. The unit outputs 1 where 2x_1-x_2-1\ge 0, the side containing the direction w=(2,-1), below the line.
- (1,0): 2-0-1=1\ge 0, output 1.
- (0,0): -1<0, output 0.
- (1,2): 2-2-1=-1<0, output 0.
The distance from the origin to the boundary is |w_0|/\lVert w\rVert=1/\sqrt{5}=0.447.