Metrics

There is more than one way to define the norm || u ||, and these correspond to different ways of measuring distance, i.e., to different metrics. Two of the most common are

Euclidean metric: || u || = sqrt( u₁² + u₂² + ... + u_d² )

Manhattan (or taxicab) metric: || u || = |u₁| + |u₂| + ... + |u_d|

In our template-matching example of classifying characters by counting the number of disagreements, we were implicitly using a Manhattan metric.* For the rest of these notes, we will use either the Euclidean distance or something called the Mahalanobis distance. We will see that

Contours of constant Euclidean distance are circles (or spheres)
Contours of constant Manhattan distance are squares (or boxes)
Contours of constant Mahalanobis distance are ellipses (or ellipsoids)

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