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Paper:

TR06-042 | 16th March 2006 00:00

Adaptive Sampling and Fast Low-Rank Matrix Approximation

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TR06-042
Authors: Amit Deshpande, Santosh Vempala
Publication: 21st March 2006 14:40
Downloads: 1439
Keywords: 


Abstract:

We prove that any real matrix $A$ contains a subset of at most
$4k/\eps + 2k \log(k+1)$ rows whose span ``contains" a matrix of
rank at most $k$ with error only $(1+\eps)$ times the error of the
best rank-$k$ approximation of $A$. This leads to an algorithm to
find such an approximation with complexity essentially
$O(Mk/\eps)$, where $M$ is the number of nonzero entries of $A$.
The algorithm maintains sparsity, and in the streaming model, it
can be implemented using only $2(k+1)(\log(k+1)+1)$ passes over
the input matrix. Previous algorithms for low-rank approximation
use only one or two passes but obtain an additive approximation.



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