A Sparse Johnson--Lindenstrauss Transform

Details A Sparse Johnson--Lindenstrauss Transform A Sparse Johnson--Lindenstrauss Transform

2010-04-23

arxiv.org/abs/1004.4240v1

Computer Science

Data Structures and Algorithms

10 pages, conference version.

Scientific paper

Dimension reduction is a key algorithmic tool with many applications including nearest-neighbor search, compressed sensing and linear algebra in the streaming model. In this work we obtain a {\em sparse} version of the fundamental tool in dimension reduction --- the Johnson--Lindenstrauss transform. Using hashing and local densification, we construct a sparse projection matrix with just $\tilde{O}(\frac{1}{\epsilon})$ non-zero entries per column. We also show a matching lower bound on the sparsity for a large class of projection matrices. Our bounds are somewhat surprising, given the known lower bounds of $\Omega(\frac{1}{\epsilon^2})$ both on the number of rows of any projection matrix and on the sparsity of projection matrices generated by natural constructions. Using this, we achieve an $\tilde{O}(\frac{1}{\epsilon})$ update time per non-zero element for a $(1\pm\epsilon)$-approximate projection, thereby substantially outperforming the $\tilde{O}(\frac{1}{\epsilon^2})$ update time required by prior approaches. A variant of our method offers the same guarantees for sparse vectors, yet its $\tilde{O}(d)$ worst case running time matches the best approach of Ailon and Liberty.

Affiliated with

Also associated with

No associations

Rating

A Sparse Johnson--Lindenstrauss Transform does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with A Sparse Johnson--Lindenstrauss Transform, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and A Sparse Johnson--Lindenstrauss Transform will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-386967