The McCulloch–Pitts unit compresses a cell to a weighted sum and a firing threshold. The perceptron is that unit with weights you are allowed to change. One unit implements a linear cut, nothing more.
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Lesson close · Neural Networks Lab
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Continue to Perceptron Math & Learning Rule, the next prerequisite step in this course.
Hypotheses. Labels used with this unit are binary. The decision boundary is the hyperplane . No probability is attached to ; the unit returns a class. A constant feature with weight is the same object as an explicit bias.
The score is a signed distance from the hyperplane, up to scaling by . The geometric margin of a point is . The threshold discards the magnitude and keeps only the side.
The biological picture (many inputs, one spike or none) explains why the map is a sum plus a cutoff. The mathematics does not use spike timing. Composition is a later step: if you feed forward with no other nonlinearity, a stack of units is still one affine map. The interesting case, a hidden layer with a nonlinear gate, is not this lesson’s unit; it is why one perceptron is only the first object.
AND and OR on binary pairs are linearly separable, so one unit can represent them. XOR is not: the positive inputs and cannot be cut from and by one line. That negative fact is a property of this definition, not of a bad step size.
A long-short rule on a vector of standardized factors is a perceptron without a learned threshold update. It is the right baseline when you believe the two return signs are linearly separable in those factors, and the wrong baseline when the signal is an interaction (the XOR pattern).
Interactive · perceptron threshold
x=1 · ŷ=sign(wx)=1
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