Optimal Sketching for Residual Error Estimation for Matrix and Vector Norms
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| Natura: | Preprint |
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2024
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| author | Li, Yi Lin, Honghao Woodruff, David P. |
| author_facet | Li, Yi Lin, Honghao Woodruff, David P. |
| contents | We study the problem of residual error estimation for matrix and vector norms using a linear sketch. Such estimates can be used, for example, to quickly assess how useful a more expensive low-rank approximation computation will be. The matrix case concerns the Frobenius norm and the task is to approximate the $k$-residual $\|A - A_k\|_F$ of the input matrix $A$ within a $(1+ε)$-factor, where $A_k$ is the optimal rank-$k$ approximation. We provide a tight bound of $Θ(k^2/ε^4)$ on the size of bilinear sketches, which have the form of a matrix product $SAT$. This improves the previous $O(k^2/ε^6)$ upper bound in (Andoni et al. SODA 2013) and gives the first non-trivial lower bound, to the best of our knowledge. In our algorithm, our sketching matrices $S$ and $T$ can both be sparse matrices, allowing for a very fast update time. We demonstrate that this gives a substantial advantage empirically, for roughly the same sketch size and accuracy as in previous work.
For the vector case, we consider the $\ell_p$-norm for $p>2$, where the task is to approximate the $k$-residual $\|x - x_k\|_p$ up to a constant factor, where $x_k$ is the optimal $k$-sparse approximation to $x$. Such vector norms are frequently studied in the data stream literature and are useful for finding frequent items or so-called heavy hitters. We establish an upper bound of $O(k^{2/p}n^{1-2/p}\operatorname{poly}(\log n))$ for constant $ε$ on the dimension of a linear sketch for this problem. Our algorithm can be extended to the $\ell_p$ sparse recovery problem with the same sketching dimension, which seems to be the first such bound for $p > 2$. We also show an $Ω(k^{2/p}n^{1-2/p})$ lower bound for the sparse recovery problem, which is tight up to a $\mathrm{poly}(\log n)$ factor. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_08494 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Optimal Sketching for Residual Error Estimation for Matrix and Vector Norms Li, Yi Lin, Honghao Woodruff, David P. Data Structures and Algorithms Machine Learning We study the problem of residual error estimation for matrix and vector norms using a linear sketch. Such estimates can be used, for example, to quickly assess how useful a more expensive low-rank approximation computation will be. The matrix case concerns the Frobenius norm and the task is to approximate the $k$-residual $\|A - A_k\|_F$ of the input matrix $A$ within a $(1+ε)$-factor, where $A_k$ is the optimal rank-$k$ approximation. We provide a tight bound of $Θ(k^2/ε^4)$ on the size of bilinear sketches, which have the form of a matrix product $SAT$. This improves the previous $O(k^2/ε^6)$ upper bound in (Andoni et al. SODA 2013) and gives the first non-trivial lower bound, to the best of our knowledge. In our algorithm, our sketching matrices $S$ and $T$ can both be sparse matrices, allowing for a very fast update time. We demonstrate that this gives a substantial advantage empirically, for roughly the same sketch size and accuracy as in previous work. For the vector case, we consider the $\ell_p$-norm for $p>2$, where the task is to approximate the $k$-residual $\|x - x_k\|_p$ up to a constant factor, where $x_k$ is the optimal $k$-sparse approximation to $x$. Such vector norms are frequently studied in the data stream literature and are useful for finding frequent items or so-called heavy hitters. We establish an upper bound of $O(k^{2/p}n^{1-2/p}\operatorname{poly}(\log n))$ for constant $ε$ on the dimension of a linear sketch for this problem. Our algorithm can be extended to the $\ell_p$ sparse recovery problem with the same sketching dimension, which seems to be the first such bound for $p > 2$. We also show an $Ω(k^{2/p}n^{1-2/p})$ lower bound for the sparse recovery problem, which is tight up to a $\mathrm{poly}(\log n)$ factor. |
| title | Optimal Sketching for Residual Error Estimation for Matrix and Vector Norms |
| topic | Data Structures and Algorithms Machine Learning |
| url | https://arxiv.org/abs/2408.08494 |