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This is to do with high dimensional geometry, which I'm always useless with. Suppose with have some large integer n and some small $\epsilon>0$. Working in the unit sphere of $\mathbb R^n$ or $\mathbb C^n$, I want to pick a large family of vectors $(u_i)_{i=1}^k$ which is almost orthogonal, in the sense that $|(u_i|u_j)| < \epsilon$ when $i\not=j$. I guess I'm interested in how the biggest choice of $k$ grows with $n$ and $\epsilon$.

For example, we can let $\{u_1,\cdots,u_n\}$ be the usual basis, and then choose $u_{n+1} = (1,1,\cdots,1)/n^{1/2}$, which works if $n^{-1/2} < \epsilon$. Then you can let $u_{n+2} = (1,\cdots,1,-1,\cdots,-1)/n^{1/2}$ and so forth, but it's not clear to me how far you can go.

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Hi Matt. I have a dim recollection of the following question rom 2nd-year undergrad homework: given $k$ almost orthogonal vectors in $R^k$ in the sense you describe, obtain a lower bound on the dimension $n$ of the subspace they span. Turning this round would seem to give an upper bound on $k$ in terms of $n$, I think? –  Yemon Choi May 16 '10 at 9:30
I've accepted Bill's answer, as it was the first to give a nice asymptotic answer. I thought that this was an innocuous question, but it's generated some nice responses. Thanks all. –  Matthew Daws May 17 '10 at 8:13

6 Answers 6

up vote 15 down vote accepted

Matt, to get $k$ points, you only need $n\ge C \epsilon^{-2} \log k$. See http://en.wikipedia.org/wiki/Johnson%E2%80%93Lindenstrauss_lemma or Google "Johnson-Lindenstrauss lemma".

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Bill, do you know anything about this constant $C$? The wiki says $8$, but do you know a paper which contains the best known one? –  Seraj May 29 '13 at 5:18

The following has always been one of my favourite facts in extremal combinatorics. If you want to pick unit vectors in R^n such that the inner product between any two of them is at most 0, then the best you can do is choose 2n vectors (an orthonormal basis and minus that basis). If you relax the condition to "at most epsilon" then you can get exponentially many by a volume argument or probabilistic methods, as other people have remarked. And if you go in the other direction, insisting that the inner product is at most -epsilon, then the biggest number of vectors you can choose is bounded above independently of n -- it's of order 1/epsilon. To prove that last fact, you calculate the norm of the sum of the vectors in two different ways -- a nice exercise I won't do here. I don't know how much is known if you impose the condition that no inner product is more than epsilon(n) for some function that tends to zero from above. It's not clear that the simple probabilistic argument gives the right result in this regime. But in general I very much like the way the function has three such different behaviours.

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@Tim: The standard volume argument using a maximal ${\sqrt 2} -\epsilon$-separated set seems not to work. What volume argument do you have in mind? –  Bill Johnson May 17 '10 at 11:43
Almost a year later I've just seen your comment. I had in mind that each point you put in rules out a spherical cap of spherical radius $\pi/2-\epsilon$. Since this has exponentially small volume, you can just greedily add exponentially many points. I think we must be talking at cross purposes though ... –  gowers May 14 '11 at 20:03
Looking at this almost a further year later, I'm still confused by Bill's remark, because what I wrote in the previous comment seems (i) correct and (ii) the standard volume argument that he discusses. Can anyone shed light on this? –  gowers Apr 3 '12 at 13:12
(four years later now) I think the confusion comes from the fact there are two different "standard volume arguments" (1) The greedy algorithm, which works primarily in the unit sphere (a "surface" argument) and (2) The argument based on the fact that a maximal delta-separated set is delta-dense, which works primarily in the unit ball (a genuine "volume" argument). In the present situation, (1) applies but (2) doesn't. –  Guillaume Aubrun Jun 14 '14 at 12:23

Indeed, what Bill Johnson wrote can hold even for points all of whose coordinates are $\pm 1/\sqrt{n}$. First choose $k = \exp(\epsilon^2 n/4)$ vectors $v_1, \dots, v_k$ by choosing each coordinate to be $\pm 1$ with probability $1/2$ each. Then define $u_i = v_i/\sqrt{n}$. A Chernoff bound shows that the probability of $|\langle u_i, u_j \rangle| \geq \epsilon$ is at most $2 \exp(-(\epsilon^2/2)n)$. This equals $2/k^2$ by choice of $k$, and hence one can take a Union Bound over the at most $\binom{k}{2} < k^2/2$ pairs $(i,j)$ to show that there's a positive probability of $|\langle u_i, u_j \rangle| < \epsilon$ holding for all $i \neq j$.

PS: Perhaps I got the constants wrong, but they can always be adjusted to make things work out.

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The Johnson-Lindenstrauss lemma states that you can have $k = 2^{\Omega(\epsilon^2 n)}$. It's also known that you cannot have $k$ larger than $2^{O(\epsilon^2 \log(1/\epsilon) n)}$ so that the Johnson-Lindenstrauss lemma gives you a near-tight answer. See the last section of "Problems and results in Extremal Combinatorics Part I" by Noga Alon for a proof of this latter fact.

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At least in the real case the buzzword is "spherical codes".

The idea is to find as large as possible a set on an $n$-sphere whose distances are at least a given amount apart. It's a spherical geometry analogue of the dense sphere-packing problem and generalizes the kissing number problem for spheres.

As with these classical problems, lots of partial results are known but rather fewer are proved to be optimal.

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"buzzword" is correct: I'd done some searching for "almost orthogonal", but found rather different problems! But this is exactly what I was after (although it seems that classically the interest is in low dimensions...) –  Matthew Daws May 16 '10 at 8:07

This was too long to be a comment, my apologies:

Further to Tim Gowers' comment and the following discussion, Chapter 6 of Bollobas' lovely white book "Combinatorics - set systems hypergraphs, families of vectors, and combinatorial probability" studies exactly this kind of question,.

In fact Tim's comment corresponds to Theorem 6 in that chapter of the book, I believe. The language of that theorem uses symmetric difference of subsets of $\{1,2,..,n\}$, which can be converted to inner products of normalized $n$ dimensional vectors with entries $\pm 1/\sqrt{n}$.

Let $\epsilon>0$ be small. The second case of Theorem 6 essentially states that if the normalized inner product is allowed to be in $[-1,\epsilon]$, equivalently, if the pairwise symmetric differences of the sets in the family are all slightly less than $n/2$ in relative terms, then the number of sets in the family, and hence the number of vectors with entries $\pm 1/\sqrt{n}$ can be as large as $2^{\epsilon n}.$

The proof of this part of the Theorem is given as an exercise with a nice hint in the book: Focus on subsets of size $k=\lceil n/2 \rceil$ and do sphere packing in the Hamming space.

The variation with the OPs question is that the allowed range for the normalized inner product is $[-\epsilon, +\epsilon]$ in the OPs question. From Jelani Nelson's answer, it seems that the penalty for restricting the inner product to a band is perhaps not as severe as one might expect. We go from $\epsilon n$ down to $\epsilon^2 \log(1/\epsilon)n$ in the exponent.

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