It's weird how stuff bubbles up from the back of your brain...

[elfs]

Last night I realized that, given you have two vectors in a 3-dimensional space starting from the same point that are not the same vector, that those two vectors create a plane-- then the same is true for any n-dimensional space. 3 dimensionas, 10 dimensions, 40 dimensions, it doesn't matter.

Why is this important? Simple; once you've identified the plane, you can find the angle between the two vectors.

If can get two people to answer 100 yes or no questions, you can create two vectors in a 100-d space and find the angle between those vectors. Those people with the smallest angle will have the greatest similarities in response. This is the basis of a vast number of recommendation engines.

It was the n-dimensionality that was bugging me. I finally grasped how little that matters in the end.

Comments

Technically, you need an inner product

[blaisepascal]

It doesn't matter the dimension of the vector space, buy you need an inner product to define the "angle". The standard Euclidean dot product usually works, but it's not the only valid inner product. Another might make sense.

In fact, the choice of inner product is probably the place to tune the recommendation engine.