使用稀疏性的快速$p$ -范数回归
Faster $p$-Norm Regression Using Sparsity
摘要：
对于矩阵$A\in \mathbb{R}^{n\times d}$与$n\geq d$，我们考虑$\min \|Ax-b\|_p^p, \, b\in \mathbb{R}^n$与$\min_{A^\top x=b} \|x\|_p^p,\, b\in \mathbb{R}^d$的对偶问题.我们提高了运行时解决这些问题的高精度为每个$p>1$足够稀疏的矩阵。我们表明，最近的进展，快速稀疏线性求解器可以利用获得比矩阵乘法算法更快的任何$p > 1$，即，在时间$\tilde{O}(pn^θ)$中，对于某些$θ< ω$，矩阵乘法常数。通过对平滑$p$ -范数函数的行采样定理，我们给出了第一个用$1 < p \leq 2$求解$\min \|Ax-b\|_p^p$的高精度输入稀疏$p$-范数回归算法.对于任何$1<p\leq 2$，该算法在时间上运行$\tilde{O}(\text{nnz}(A) + d^4)$，对于接近$2$的$p$，该算法在时间上运行$\tilde{O}(\text{nnz}(A) + d^θ)$，改进了先前的最佳界限，其中$d$的指数随着$\max\{p, p/(p-1)\}$增长。
英文摘要：
For a matrix $A\in \mathbb{R}^{n\times d}$ with $n\geq d$, we consider the dual problems of $\min \|Ax-b\|_p^p, \, b\in \mathbb{R}^n$ and $\min_{A^\top x=b} \|x\|_p^p,\, b\in \mathbb{R}^d$. We improve the runtimes for solving these problems to high accuracy for every $p>1$ for sufficiently sparse matrices. We show that recent progress on fast sparse linear solvers can be leveraged to obtain faster than matrix-multiplication algorithms for any $p > 1$, i.e., in time $\tilde{O}(pn^θ)$ for some $θ< ω$, the matrix multiplication constant. We give the first high-accuracy input sparsity $p$-norm regression algorithm for solving $\min \|Ax-b\|_p^p$ with $1 < p \leq 2$, via a new row sampling theorem for the smoothed $p$-norm function. This algorithm runs in time $\tilde{O}(\text{nnz}(A) + d^4)$ for any $1<p\leq 2$, and in time $\tilde{O}(\text{nnz}(A) + d^θ)$ for $p$ close to $2$, improving on the previous best bound where the exponent of $d$ grows with $\max\{p, p/(p-1)\}$.
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