A Sparse Johnson--lindenstrauss Transform
Dasgupta Anirban, Kumar Ravi, Sarlós Tamás. Arxiv 2010
[Paper]
ARXIV
Independent
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 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 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 update time per
non-zero element for a -approximate projection, thereby
substantially outperforming the update time
required by prior approaches. A variant of our method offers the same
guarantees for sparse vectors, yet its worst case running time
matches the best approach of Ailon and Liberty.
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