Lesson 7 gave you logistic regression: a linear model — it computes a weighted sum of the input features (plus a bias) and passes it through a squashing function. Geometrically, the boundary between “predict class 0” and “predict class 1” is always a straight line (or, in higher dimensions, a flat plane/hyperplane). That’s true no matter how you tune the weights — the shape of what a linear model can draw is fixed; only its position and angle are adjustable.
The four points. Consider two binary input features, x1 and x2, each either 0 or 1, and a
target label that’s the XOR of them (1 if exactly one input is 1, 0 otherwise — “different
inputs → true, same inputs → false”):
| x1 | x2 | label |
|---|---|---|
| 0 | 0 | 0 |
| 0 | 1 | 1 |
| 1 | 0 | 1 |
| 1 | 1 | 0 |
Plot these as four points on a 2D grid: (0,0) and (1,1) are label 0; (0,1) and (1,0) are label 1. The two label-0 points sit on one diagonal, the two label-1 points sit on the other diagonal — they’re interleaved, not grouped into two separable clusters.
Your task: can any single straight line, positioned and angled however you like, put both label-0 points on one side and both label-1 points on the other? Think about it geometrically before picking an answer — try sketching the four points and attempting to draw a separating line.