Vectors, Distance and Similarity
Everything Is Far Away and Nothing Is Nearby
Last timePutting Everything on the Sphere
In a space with hundreds of coordinates, distances between unrelated points stop varying much and almost every pair of directions is close to perpendicular. Both facts have consequences.
Everything so far has been arithmetic that works the same at any number of
coordinates. This lesson is about what changes anyway, and it changes because of
how sums of many small quantities behave rather than because of anything about
vectors.
Distances stop being different from each other
Take two points whose coordinates are unrelated. Their distance is built from one
small contribution per coordinate. With three coordinates a single unusual one can
dominate and distances vary a lot. With three hundred, each contributes a small
share of a large total, and the totals for different pairs come out close
together.
This is the fact behind the phrase that everything is far away in high
dimensions. More precisely: everything is about equally far away, which is worse,
because it is the differences between distances that a search relies on.
The lesson stops here
6 more paragraphs to go
You have read the opening. The rest of the argument, the problems that check whether it landed, and the lines worth keeping at the end all come with a plan.
The first lesson of every course in the library reads the whole way through, free, so you can see exactly what the rest of them are.
See the planThe contentsThis is the reading half
Starting the course gives you your own copy of it. Every idea on every page has problems standing under it, marked with a reason rather than a tick, and any sentence you do not believe can be opened and argued with. None of that can happen on a page nobody owns.
The contents