What exactly is a norm? Is it a method to calculate length/distance/magnitude of a vector? Why are there multiple ways (L-1,L-2,L-P) to calculate the length/distance/magnitude of a vector ?
I can’t seem to find an easy to understand explanation online. Please explain.
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Yes, a norm is essentially an abstract measure, a generalization of distance. A norm has to satisfy a couple important properties:
1. It has to give a positive result
2. The distance between a point and itself must be 0.
3. Scaling must be linear.
4. There may not be a shorter way between two points than the direct way (the so-called triangle inequality).
One famous norm is the taxicab metric, also known as Manhattan distance. Imagine you’re in the city of manhattan and you want to know the distance between two locations. One way to measure that distance is by air-distance, this would be the one we’re most familiar with. But you can’t just move along that path, you have to walk along the grid-layout streets. That distance is the Manhattan distance.
Note that the direct way under Manhattan distance is not as the crow flies, as the crow doesn’t adhere to it.
The great thing about norms is that you can use them for a lot of things you wouldn’t immediately associate with distances. Take for example the words “robes” and “home”. How many letter would you have to edit (insertion, deletion, substition) in order to transform one of those words into the other? You’d need 3 edits – replace r with h, b with m, and remove the s. The norm indicating this edit distance between texts is called the Levenshtein distance.
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