Sparser Johnson-Lindenstrauss Transforms

Details Sparser Johnson-Lindenstrauss Transforms Sparser Johnson-Lindenstrauss Transforms

2010-12-07

arxiv.org/abs/1012.1577v3

Computer Science

Data Structures and Algorithms

v3: proof of Theorem 25 in v2 was written incorrectly, now fixed; v2: Added another construction achieving same upper bound, a

Scientific paper

We give two different Johnson-Lindenstrauss distributions, each with column sparsity s = Theta(eps^{-1}log(1/delta)) and embedding into optimal dimension k = O(eps^{-2}log(1/delta)) to achieve distortion 1+/-eps with probability 1-delta. That is, only an O(eps)-fraction of entries are non-zero in each embedding matrix in the supports of our distributions. These are the first distributions to provide o(k) sparsity for all values of eps,delta. Previously the best known construction obtained s = Theta (eps^{-1}log^2(1/delta)). One of our distributions can be sampled from using O(log(1/delta)log d) random bits. Some applications that use Johnson-Lindenstrauss embeddings as a black box, such as those in approximate numerical linear algebra ([Sarlos, FOCS 2006], [Clarkson-Woodruff, STOC 2009]), require exponentially small delta. Our linear dependence on log(1/delta) in the sparsity is thus crucial in these applications to obtain speedup.

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