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
Nearby in the stack
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 Vybíral, 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 Vybíral 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