Vector size reduction preserving dot product

Asked Jul 16 '22 at 20:24

Active Jul 17 '22 at 15:19

Viewed 108 times

I have Vectors with size n=1000..3000 which have a mean=0 and std-dev=1/sqrt(n) and length/norm=1.

My question is what formula, algorithm I can use to reduce the vectors size but preserve the DOT product similarity i.e. after re-scaling the dot product between any 2 vectors should be approximately the same as the original vectors ?

random selection seems to work:

% : normalizes the operands and does the DOT product..

In [34]: ab % bc
Out[34]: 0.445

In [35]: ab[::2] % bc[::2]
Out[35]: 0.424

In [36]: ab[::3] % bc[::3]
Out[36]: 0.440

In [39]: rnd = np.random.randint(0,1000,300)

In [40]: ab[rnd] % bc[rnd]
Out[40]: 0.450

edited Jul 17 '22 at 15:19

asked Jul 16 '22 at 20:24

sten

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You want to reduce the vector dimension, or the vector length? – Colim Jul 16 '22 at 21:29
f.e. from n=1000 to n=500 – sten Jul 16 '22 at 22:35
This can’t be done independently of the vector values, since any continuous projection will cause some very different vectors to become identical. – Davis Herring Jul 17 '22 at 00:08
what if i have all the vectors – sten Jul 17 '22 at 01:30
@sten: Do you have more vectors than dimensions? – Davis Herring Jul 17 '22 at 03:26
10 000 to 100 000 vectors – sten Jul 17 '22 at 14:54
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(If you don’t use `@`, your reply may go unnoticed.) This is impossible in general: consider reducing the 3 3D elementary unit vectors to 2D, where the most orthogonal 3 vectors can be is -1/2. Are you intending to rely on more convenient vectors than that? – Davis Herring Jul 17 '22 at 15:47
Let us [continue this discussion in chat](https://chat.stackoverflow.com/rooms/246513/discussion-between-sten-and-davis-herring). – sten Jul 17 '22 at 17:59

Vector size reduction preserving dot product

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