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Let be a natural number. We consider the question of how many “almost orthogonal” unit vectors one can place in the Euclidean space . Of course, if we insist on being exactly orthogonal, so that for all distinct , then we can only pack at most unit vectors into this space. However, if one is willing to relax the orthogonality condition a little, so that is small rather than zero, then one can pack a lot more unit vectors into , due to the important fact that pairs of vectors in high dimensions are typically almost orthogonal to each other. For instance, if one chooses uniformly and independently at random on the unit sphere, then a standard computation (based on viewing the as gaussian vectors projected onto the unit sphere) shows that each inner product concentrates around the origin with standard deviation and with gaussian tails, and a simple application of the union bound then shows that for any fixed , one can pack unit vectors into whose inner products are all of size .
One can remove the logarithm by using some number theoretic constructions. For instance, if is twice a prime , one can identify with the space of complex-valued functions , whee is the field of elements, and if one then considers the different quadratic phases for , where is the standard character on , then a standard application of Gauss sum estimates reveals that these unit vectors in all have inner products of magnitude at most with each other. More generally, if we take and consider the different polynomial phases for , then an application of the Weil conjectures for curves, proven by Weil, shows that the inner products of the associated unit vectors with each other have magnitude at most .
As it turns out, this construction is close to optimal, in that there is a polynomial limit to how many unit vectors one can pack into with an inner product of :
In particular, for fixed and large , the number of unit vectors one can pack in whose inner products all have magnitude at most will be . This doesn’t quite match the construction coming from the Weil conjectures, although it is worth noting that the upper bound of for the inner product is usually not sharp (the inner product is actually times the sum of unit phases which one expects (cf. the Sato-Tate conjecture) to be uniformly distributed on the unit circle, and so the typical inner product is actually closer to ).
Note that for , the case of the above theorem (or more precisely, Lemma 2 below) gives the bound , which is essentially optimal as the example of an orthonormal basis shows. For , the condition is trivially true from Cauchy-Schwarz, and can be arbitrariy large. Finally, in the range , we can use a volume packing argument: we have , so of we set , then the open balls of radius around each are disjoint, while all lying in a ball of radius , giving rise to the bound for some absolute constant .
As I learned recently from Philippe Michel, a more precise version of this theorem is due to Kabatjanskii and Levenstein, who studied the closely related problem of sphere packing (or more precisely, cap packing) in the unit sphere of . However, I found a short proof of the above theorem which relies on one of my favorite tricks – the tensor power trick – so I thought I would give it here.
We begin with an easy case, basically the case of the above theorem:
Proof: Suppose for contradiction that . We consider the Gram matrix . This matrix is real symmetric with rank at most , thus if one subtracts off the identity matrix, it has an eigenvalue of with multiplicity at least . Taking Hilbert-Schmidt norms, we conclude that
But by hypothesis, the left-hand side is at most , giving the desired contradiction.
To amplify the above lemma to cover larger values of , we apply the tensor power trick. A direct application of the tensor power trick does not gain very much; however one can do a lot better by using the symmetric tensor power rather than the raw tensor power. This gives
Corollary 3 Let be a natural number, and let be unit vectors in such that for all distinct . Then .
Proof: We work in the symmetric component of the tensor power , which has dimension . Applying the previous lemma to the tensor powers , we obtain the claim.
Using the trivial bound , we can lower bound
We can thus prove Theorem 1 by setting for some sufficiently large absolute constant .
As many readers may already know, my good friend and fellow mathematical blogger Tim Gowers, having wrapped up work on the Princeton Companion to Mathematics (which I believe is now in press), has begun another mathematical initiative, namely a “Tricks Wiki” to act as a repository for mathematical tricks and techniques. Tim has already started the ball rolling with several seed articles on his own blog, and asked me to also contribute some articles. (As I understand it, these articles will be migrated to the Wiki in a few months, once it is fully set up, and then they will evolve with edits and contributions by anyone who wishes to pitch in, in the spirit of Wikipedia; in particular, articles are not intended to be permanently authored or signed by any single contributor.)
So today I’d like to start by extracting some material from an old post of mine on “Amplification, arbitrage, and the tensor power trick” (as well as from some of the comments), and converting it to the Tricks Wiki format, while also taking the opportunity to add a few more examples.
Title: The tensor power trick
Quick description: If one wants to prove an inequality for some non-negative quantities X, Y, but can only see how to prove a quasi-inequality that loses a multiplicative constant C, then try to replace all objects involved in the problem by “tensor powers” of themselves and apply the quasi-inequality to those powers. If all goes well, one can show that for all , with a constant C which is independent of M, which implies that as desired by taking roots and then taking limits as .