Journal of Computer and System Sciences 66 (2003) 671687
Database-friendly random projections:
Johnson-Lindenstrauss with binary coins
Microsoft Research, One Microsoft Way, Redmond, WA 98052, USA
Received 28 August 2001; revised 19 July 2002
A classic result of Johnson and Lindenstrauss asserts that any set of n points in d-dimensional Euclidean
space can be embedded into k-dimensional Euclidean space--where k is logarithmic in n and independent
of d--so that all pairwise distances are maintained within an arbitrarily small factor. All known
constructions of such embeddings involve projecting the n points onto a spherically random k-dimensional
hyperplane through the origin. We give two constructions of such embeddings with the property that all
elements of the projection matrix belong in fÀ1; 0; þ1g: Such constructions are particularly well suited for
database environments, as the computation of the embedding reduces to evaluating a single aggregate over
k random partitions of the attributes.
r 2003 Elsevier Science (USA). All rights reserved.
Consider projecting the points of your favorite sculpture first onto the plane and then onto a
single line. The result amply demonstrates the power of dimensionality.