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It seems, from a quick (and incomprehensive) survey of the literature (e.g. N. Ailon, B. Chazelle, SIAM J. Comput. 39 (2009), 302-322.) that there are many fast algorithms (and constructions) for the Johnson-Lindenstrauss transform: $f \in \mathbb{R}^{d} \to \mathbb{R}^{k}$, for $n$ points in $\mathbb{R}^d$ and $k \in O(\log n/\epsilon)$. I'm interested in computing the inverse of J-L transform. I assume it is not unique in general. Are there any references I should be reading on the inverse transform of J-L?

If I understand correctly, the FJLT transform (sparse times Hadamard times diagonal) can be represented by a matrix $R = PHD \in \mathbb{R}^{d \times k}$. Note, I'm pretty much only interested in realizations of J-L transforms which are represented by matrices. Naively, I'd think that the inverse is some sort of Moore-Penrose pseudoinverse of $R$. Is this actually what people do?

(I wasn't exactly sure if this question is suitable for cstheory.SE, so I decide posting it here.)

Raphael
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  • I am not sure which subfield of CS this belongs to; please try and choose appropriate tags. – Raphael Apr 12 '13 at 07:20
  • What are you asking exactly? I have never seen an inverse of JL being used. The problem is that the inverse would map points to affine spaces of quite large dimension. But if you wished you could compute it from the pseudoinverse – Sasho Nikolov Apr 14 '13 at 01:21

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