The Johnson-Lindenstrauss lemma is one of the corner stone results in
dimensionality reduction. It says that given N, for any set of N vectors X⊂Rn, there exists a mapping f:X→Rm such
that f(X) preserves all pairwise distances between vectors in X to within
(1±ε) if m=O(ε−2lgN). Much effort has gone
into developing fast embedding algorithms, with the Fast Johnson-Lindenstrauss
transform of Ailon and Chazelle being one of the most well-known techniques.
The current fastest algorithm that yields the optimal m=O(ε−2lgN) dimensions has an embedding time of O(nlgn+ε−2lg3N). An exciting approach towards improving this, due to
Hinrichs and Vyb\'iral, is to use a random m×n Toeplitz matrix for the
embedding. Using Fast Fourier Transform, the embedding of a vector can then be
computed in O(nlgm) time. The big question is of course whether m=O(ε−2lgN) dimensions suffice for this technique. If so, this
would end a decades long quest to obtain faster and faster
Johnson-Lindenstrauss transforms. The current best analysis of the embedding of
Hinrichs and Vyb\'iral shows that m=O(ε−2lg2N) dimensions
suffices. The main result of this paper, is a proof that this analysis
unfortunately cannot be tightened any further, i.e., there exists a set of N
vectors requiring m=Ω(ε−2lg2N) for the Toeplitz
approach to work