
Why is it a known fact that the angle between two random vectors in a high dimensional space is atleast 45 degree?
I ran the simulations with two random vectors of length 100, and they were indeed atleast 45 degrees apart. But I can't seem to understand why?
My approach:
- Take two vectors of length 100. Randomly allocate values to each component.
- Compute inverse cosine of ( a . b / |a| |b| )
The histogram is centered around 45 degrees. but why?


The curse of dimensionality suggests that any two random variables are likely to be orthogonal i.e. 90 degrees. This is often attributed to the sparsity of the space.
This then means that the maximal angle between all vectors in high dimensional space is atleast 45 degrees, which is half of 90 degrees. This is because there is a constraint on it having only positive elements.

Looking at the last question, who even uses R? Interesting question tho.
Eyeballing it 99% Frobenius norm should be captured within first 15 singular vectors from SVD. This is because adjacent cells are correlated so it would ideally capture most of that pattern.
We used SVD for minimising storage of petabytes of tabular data at Goldman.

I learnt something new today. High dimensional vectors are with length greater than 3 that can be used to represent multi-latent factors in a database.

This comment added no value. Everyone knows what a high dimensional vector is lmao.
It is not a good definition. A vector is essentially an object with a direction as well as magnitude. A one dimensional vector is a vector that exists on one of the principal axis. A two dimensional vector is a vector that is a sum of two magnitude components along x and y axis.
Similarly, a three dimensional vector is a vector that is a sum of three magnitude components along x, y and z axis.
So a high dimensional vector is a vector that is a sum of n(where n> 3) magnitude components along x,y,z … till nth principal axis.

Kuch samjh nahi aaya but Achcha laga.

Which course is this?

Aayein? Yeh konsi lipi hai?