Notice that the **x**'**x** term is the same for every class, i.e.,
for every k. To find the template **m**_{k} that minimizes
|| **x** - **m**_{k} ||, it is sufficient
to find the **m**_{k} that maximizes the bracketed
expression, **m**_{k}' **x** - 0.5 **m**_{k}'
**m**_{k}. Let us define the **linear discriminant
function** g(**x**) by

Then we can say that a minimum-Euclidean-distance clasifier classifies
an input feature vector **x** by computing c linear discriminant
functions g_{1}(**x**), g_{2}(**x**),
... , g_{c}(**x**) and assigning **x**
to the class corresponding to the maximum discriminant function. We can
also think of the linear discriminant functions as measuring the correlation
between **x** and **m**_{k}, with the
addition of a correction for the "template energy" represented
by || **m**_{k} ||^{2}. With this correction
included, a minimum-Euclidean-distance classifier is equivalent to a maximum-correlation
classifier.

Back to Inner Prod. On to Boundaries Up to Simple Classifiers