On Using Toeplitz and Circulant Matrices for Johnson-Lindenstrauss Transforms

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 \subset R^n, there exists a mapping f : X --> R^m such that f(X) preserves all pairwise distances between vectors in X to within(1 ± \eps) if m = O(\eps^{-2} lg N). 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(\eps{-2}lg N) dimensions has an embedding time of O(n lg n + \eps^{-2} lg^3 N). An exciting approach towards improving this, due to Hinrichs and Vybíral, is to use a random m times n Toeplitz matrix for the embedding. Using Fast Fourier Transform, the embedding of a vector can then be computed in O(n lg m) time. The big question is of course whether m = O(\eps^{-2} lg N) 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(\eps^{-2} lg^2 N) dimensions suffice. 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 = \Omega(\eps^{-2} lg^2 N) for the Toeplitz approach to work.