结构化输入的快速回归
Fast Regression for Structured Inputs
摘要：
我们研究$\ell_p$回归问题，这需要找到$\mathbf{x}\in\mathbb R^{d}$，最小化$\|\mathbf{A}\mathbf{x}-\mathbf{b}\|_p$的矩阵$\mathbf{A}\in\mathbb R^{n \times d}$和响应向量$\mathbf{b}\in\mathbb R^{n}$。最近有兴趣在开发子采样方法，可以优于标准技术时，$n$是非常大的这个问题。然而，所有已知的子采样方法具有指数地依赖于$p$（通常为$d^{\mathcal{O}(p)}$）的运行时间，这可能是非常昂贵的。我们改进这项工作表明，对于一大类常见的\emph{structured matrices}，如低秩矩阵，稀疏矩阵和范德蒙矩阵的组合，有基于子采样的方法$\ell_p$回归多项式依赖于$p$。例如，我们给出了一个在时间上运行的Vandermonde矩阵的$\ell_p$回归算法$\mathcal{O}(n\log^3 n+(dp^2)^{0.5+ω}\cdot\text{polylog}\,n)$，其中$ω$是矩阵乘法的指数。多项式依赖$p$至关重要地允许我们的算法自然地扩展到有效的算法$\ell_\infty$回归，通过近似的$\ell_\infty$的$\ell_{\mathcal{O}(\log n)}$。实际利益，我们还开发了一个新的子采样算法$\ell_p$回归任意矩阵，这是简单的比以前的方法$p \ge 4$。
英文摘要：
We study the $\ell_p$ regression problem, which requires finding $\mathbf{x}\in\mathbb R^{d}$ that minimizes $\|\mathbf{A}\mathbf{x}-\mathbf{b}\|_p$ for a matrix $\mathbf{A}\in\mathbb R^{n \times d}$ and response vector $\mathbf{b}\in\mathbb R^{n}$. There has been recent interest in developing subsampling methods for this problem that can outperform standard techniques when $n$ is very large. However, all known subsampling approaches have run time that depends exponentially on $p$, typically, $d^{\mathcal{O}(p)}$, which can be prohibitively expensive. We improve on this work by showing that for a large class of common \emph{structured matrices}, such as combinations of low-rank matrices, sparse matrices, and Vandermonde matrices, there are subsampling based methods for $\ell_p$ regression that depend polynomially on $p$. For example, we give an algorithm for $\ell_p$ regression on Vandermonde matrices that runs in time $\mathcal{O}(n\log^3 n+(dp^2)^{0.5+ω}\cdot\text{polylog}\,n)$, where $ω$ is the exponent of matrix multiplication. The polynomial dependence on $p$ crucially allows our algorithms to extend naturally to efficient algorithms for $\ell_\infty$ regression, via approximation of $\ell_\infty$ by $\ell_{\mathcal{O}(\log n)}$. Of practical interest, we also develop a new subsampling algorithm for $\ell_p$ regression for arbitrary matrices, which is simpler than previous approaches for $p \ge 4$.
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