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.)
Computing the inverse of Johnson-Lindenstrauss transform
reference request