eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2017-12-07
32:1
32:12
10.4230/LIPIcs.ISAAC.2017.32
article
On Using Toeplitz and Circulant Matrices for Johnson-Lindenstrauss Transforms
Freksen, Casper Benjamin
Larsen, Kasper Green
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.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol092-isaac2017/LIPIcs.ISAAC.2017.32/LIPIcs.ISAAC.2017.32.pdf
dimensionality reduction
Johnson-Lindenstrauss
Toeplitz matrices