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so, fun fact, turns out you can fit exponentially many (normalized) vectors in a list such that the pairwise inner product of any two distinct vectors is ≤ eps

In fact for small eps, you can do slightly better than the random construction via Reed-Solomon codes, see e.g. arxiv.org/pdf/1206.5725

So one angle for intuition here is good error correcting codes exist => you can find lots of nearly orthogonal vectors

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The user is asking about the connection between the original quote about fitting exponentially many nearly orthogonal vectors and the paper they linked about Reed-Solomon codes for deterministic sketching. The original comment said: "so, fun fact, turns out you can fit exponentially many (normalized) vectors in a list such that the pairwise inner product of any two distinct vectors is ≤ eps" And then mentioned that Reed-Solomon codes can do slightly better than the random construction. Looking at the paper, the key connection is in the section "Incoherent matrices from codes" where they explain how error-correcting codes can be used to construct matrices with nearly orthogonal columns. Specifically, they show: 1. You can take a code with alphabet size q, block length t, and relative distance 1-ε 2. From this code, you can construct a matrix where each column has unit ℓ2 norm and the inner product between any two distinct columns is at most ε 3. Reed-Solomon codes provide a particularly efficient construction Let me explain this connection clearly.

The paper shows exactly how the quoted comment works through the construction of "incoherent matrices" - matrices where all columns have unit 2\ell_2 norm and pairwise inner products are at most \eps\eps.

Here's the key construction from the paper:

Given an error-correcting code C={C1,,Cn}\mathcal{C} = \{C_1, \ldots, C_n\} with:

  • Alphabet size qq
  • Block length tt
  • Relative distance 1\eps1-\eps (meaning any two codewords differ in at least (1\eps)t(1-\eps)t positions)

You can build an m×nm \times n matrix AA with m=qtm = qt rows as follows:

  • Each column AiA_i corresponds to codeword CiC_i
  • Set (Ai)j,k=1/t(A_i)_{j,k} = 1/\sqrt{t} if the jj-th symbol of CiC_i equals kk, and 00 otherwise

This gives you nn columns (vectors) where:

  • Each column has exactly tt non-zero entries of value 1/t1/\sqrt{t}, so Ai2=1\|A_i\|_2 = 1
  • For distinct columns iji \neq j: Ai,Aj=tΔ(Ci,Cj)t\eps\langle A_i, A_j \rangle = \frac{t - \Delta(C_i, C_j)}{t} \leq \eps

where Δ(Ci,Cj)\Delta(C_i, C_j) is the Hamming distance between codewords.

For Reed-Solomon codes specifically, they take:

  • q=Θ(\eps1logn/(loglogn+log(1/\eps)))q = \Theta(\eps^{-1}\log n/(\log\log n + \log(1/\eps)))
  • Each codeword corresponds to evaluations of a degree-dd polynomial over Fq\mathbb{F}_q
  • This gives n>qdn > q^d codewords (exponentially many in dd)

The Reed-Solomon construction achieves m=O(\eps2(logn/(loglogn+log(1/\eps)))2)m = O(\eps^{-2}(\log n/(\log\log n + \log(1/\eps)))^2) rows, which beats random constructions when \eps<2Ω(logn)\eps < 2^{-\Omega(\sqrt{\log n})}.

So the "fun fact" is realized through error-correcting codes: good codes with large distance naturally give you many vectors with small pairwise inner products!

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