10 Jonathan Richard Shewchuk
Theorem: Let x, y be 2 points that lie on H. Then w · (y − x) = 0.
Proof: w · (y − x) = −α − (−α) = 0.
w is called the normal vector
of H,
because (as the theorem shows) w is normal (perpendicular) to H.
(I.e. w is normal to every pair of points in H.)
w
w · x = −2
w · x = 1
w · x = 0
[Draw black part first, then red parts. hyperplane.pdf ]
If w is a unit vector, then w · x + α is the signed distance from x to H.
I.e. positive on one side of H; negative on other side.
Moreover, the distance from H to the origin is α. [How do we know that?]
Hence α = 0 if and only if H passes through origin.
[w does not have to be a unit vector for the classifier to work.
If w is not a unit vector, w · x + α is the signed distance times some real.
If you want to fix that, you can rescale the equation
by computing |w| and dividing both w and α by |w|.]
The coefficients in w, plus α, are called weights or sometimes regression coefficients.
[That’s why I named the vector w; “w” stands for “weights.”]
The input data is linearly separable if there exists a hyperplane that separates all the sample points in class
C from all the points NOT in class C.
[At the beginning of this lecture, I showed you one plot that’s linearly separable and two that are not.]
[We will investigate some linear classifiers that only work for linearly separable data, then we’ll move on
to more sophisticated linear classifiers that do a decent job with non-separable data. Obviously, if your data
is not linearly separable, a linear classifier cannot do a perfect job. But we’re still happy if we can find a
classifier that usually predicts correctly.]
A Simple Classifier
Centroid method: compute mean µ
C
of all vectors in class C and mean µ
X
of all vectors NOT in C.
We use the decision function
f (x) = (µ
C
− µ
X
)
| {z }
normal vector
·x − (µ
C
− µ
X
) ·
µ
C
+ µ
X
2
| {z }
midpoint between µ
C
, µ
X
so that decision boundary is the hyperplane that bisects line segment w/endpoints µ
C
, µ
X
.
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