Das XOR-Problem: Warum ein Neuron nicht genügt

The XOR rule outputs 1 when exactly one input is 1, and 0 otherwise. It looks simple, yet it broke early neural networks. 🤔

XOR gives 0 for (0,0) and (1,1), but 1 for (0,1) and (1,0). The matching pairs say no, the mismatched pairs say yes.

Place the four inputs on a grid. The two yes points sit on opposite corners, with the no points filling the other two.

Try to split the yes corners from the no corners with one straight line. You simply cannot do it. The classes interlock diagonally.

XOR is the classic example of data that is not linearly separable. A single neuron, limited to a straight boundary, can never solve it.

Train a perceptron on XOR and it never converges. The weights keep wobbling because no line satisfies all four examples at once.

This XOR limit, exposed in 1969, helped trigger an AI winter. People doubted neural nets could ever handle hard problems.

The fix is to stack neurons. A hidden layer lets the network combine several lines into a more complex shape.

With a hidden layer plus nonlinearity, the network can bend its boundary. Now it wraps around the yes corners and ignores the no ones.

Just two hidden neurons feeding one output neuron can crack XOR. Each hidden unit draws a line, and the output combines them.

XOR is the smallest proof that depth matters. Stacking layers is what lets networks learn curves, corners, and real-world complexity.

Recall why XOR defeats a single neuron.

XOR is not linearly separable, so one neuron fails. Adding a hidden layer lets the network bend its boundary and solve it. Depth unlocks power. 🚀