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emil-lp

Favorite lemmata:

Johnson–Lindenstrauss lemma

https://en.wikipedia.org/wiki/Johnson%E2%80%93Lindenstrauss_...

Isolation lemma

https://en.wikipedia.org/wiki/Isolation_lemma

Schwartz–Zippel lemma

https://en.wikipedia.org/wiki/Schwartz%E2%80%93Zippel_lemma

Lovász local lemma

https://en.wikipedia.org/wiki/Lov%C3%A1sz_local_lemma

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The division of lemmas and theorems is really a bit artificial for these things. But yeah I think the spirit is that a theorem is an object that you aim to study, while a lemma is something you use to do that. Fermat's last theorem was a target, but the techniques including lemmas used and developed for it are the real prize for a working mathematician. Sculptures are kind of the point, but there's no question the tools used for sculpting are more useful and "worth" more in that sense.

Ah the JL lemma. Probably one of my favorite too. I'm teaching a mathematics of data course next semester, and even though we don't assume probability as a prerequisite I'm going to find a way to talk about that idea.

I am sure you know this, but inner-products are not preserved, the el_2 distance is (approximately).

So it needs judicious care when used along with algorithm s that work with inner-products

If you preserve the l2 distance you preserve the inner product, that's somewhat tautological in an L2 space. Just that the degree you can preserve inner products can be misleading, main problem is that orthogonal vectors may only become near-orthogonal which is sometimes a big deal, though perfect correlations are preserved because the JL transform is linear. Both can be seen looking at: https://en.wikipedia.org/wiki/Polarization_identity

> If you preserve the l2 distance you preserve the inner product

That's trivially untrue. You can move the origin around and that doesn't change the el_2 metric but will change the inner product.

This would not happen for random rotations of course because they do not change the origin. However random Euclidean motions can change the origin.

Right, indeed you need to first preserve the origin, but also that is trivially true for a linear map like JL.

As far as I can recall JL holds for affine transformations too, in any case it's an existence result. Have to double check on the affine bit.

The popular proof does uses random linear transforms and they indeed will not change the origin, but that's just one class of transforms with the JL property.

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