ECCC-Report TR06-042https://eccc.weizmann.ac.il/report/2006/042Comments and Revisions published for TR06-042en-usTue, 21 Mar 2006 14:40:24 +0200
Paper TR06-042
| Adaptive Sampling and Fast Low-Rank Matrix Approximation |
Amit Deshpande,
Santosh Vempala
https://eccc.weizmann.ac.il/report/2006/042We 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.
Tue, 21 Mar 2006 14:40:24 +0200https://eccc.weizmann.ac.il/report/2006/042