DizzyPretzel
DizzyPretzel

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:

  1. Take two vectors of length 100. Randomly allocate values to each component.
  2. Compute inverse cosine of ( a . b / |a| |b| )

The histogram is centered around 45 degrees. but why?

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12mo ago
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SwirlyPretzel
SwirlyPretzel
Google12mo

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.

ZestyHamster
ZestyHamster
Google12mo

This is peak Grapevine moment when you getting an assignment solved from someone at Google😂

GigglyUnicorn
GigglyUnicorn
Dezerv12mo

@SarjapurDon what will you do with that info? It’s so random. xD

JumpyTaco
JumpyTaco

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.

FuzzyPretzel
FuzzyPretzel

Can we connect in DMs? I have a portfolio company building in this space.

GoofyHamster
GoofyHamster
Swiggy12mo

I have some experience with this too. I have DM you. Please respond.

GroovyKoala
GroovyKoala
Adobe12mo

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.

SparklyDonut
SparklyDonut
Apple12mo

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.

DizzySushi
DizzySushi

Kuch samjh nahi aaya but Achcha laga.

PrancingCupcake
PrancingCupcake

Which course is this?

GroovyKoala
GroovyKoala
Eviden12mo

Aayein? Yeh konsi lipi hai?

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