920 research outputs found

    Optimal Column-Based Low-Rank Matrix Reconstruction

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    We prove that for any real-valued matrix X∈RmΓ—nX \in \R^{m \times n}, and positive integers rβ‰₯kr \ge k, there is a subset of rr columns of XX such that projecting XX onto their span gives a r+1rβˆ’k+1\sqrt{\frac{r+1}{r-k+1}}-approximation to best rank-kk approximation of XX in Frobenius norm. We show that the trade-off we achieve between the number of columns and the approximation ratio is optimal up to lower order terms. Furthermore, there is a deterministic algorithm to find such a subset of columns that runs in O(rnmΟ‰log⁑m)O(r n m^{\omega} \log m) arithmetic operations where Ο‰\omega is the exponent of matrix multiplication. We also give a faster randomized algorithm that runs in O(rnm2)O(r n m^2) arithmetic operations.Comment: 8 page
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