强多项式帧缩放至高精度
Strongly Polynomial Frame Scaling to High Precision
AI总结：
本文提出一种强多项式确定性算法，以对数收敛速度解决帧缩放问题，将迭代次数从O(n^5 d^11/ε^5)降至O(n^3 log(n/ε))，并改进矩阵缩放的迭代界，同时给出解大小的新界。
摘要：
帧缩放问题是：给定向量 $U := \{u_{1}, ..., u_{n} \} \subseteq \mathbb{R}^{d}$，边际 $c \in \mathbb{R}^{n}_{++}$，以及精度 $\varepsilon > 0$，寻找左缩放和右缩放 $L \in \mathbb{R}^{d \times d}, r \in \mathbb{R}^n$，使得 $(v_1,\dots,v_n) := (Lu_1 r_1,\dots,Lu_nr_n)$ 同时满足 $\sum_{i=1}^n v_i v_i^{\mathsf{T}} = I_d$ 和 $\\|v_{j}\\|_{2}^{2} = c_{j}, \forall j \in [n]$，误差不超过 $\varepsilon$。这个问题在线性代数和计算机科学的各个领域中出现。在这项工作中，我们给出了一个具有 $\log(1/\varepsilon)$ 收敛速度的帧缩放强多项式算法。这回答了 Diakonikolas、Tzamos 和 Kane（STOC 2023）提出的一个问题，他们给出了第一个具有 poly$(1/\varepsilon)$ 收敛速度的强多项式随机算法，针对特殊情况 $c = \frac{d}{n} 1_{n}$。我们的算法是确定性的，适用于一般的 $c \in \mathbb{R}^{n}_{++}$，并且需要 $O(n^{3} \log(n/\varepsilon))$ 次迭代，而 DTK 需要 $O(n^{5} d^{11}/\varepsilon^{5})$ 次迭代。通过将 Linial、Samorodnitsky 和 Wigderson（Combinatorica 2000）用于矩阵缩放的框架提升到帧，我们能够简化算法和分析。我们的主要技术贡献是将 LSW 的势分析推广到帧设置，并在强多项式时间内计算一个更新步骤，该步骤在每次迭代中实现几何进展。事实上，我们可以调整我们的结果以给出强多项式矩阵缩放的改进分析，将 LSW 的 $O(n^{5} \log(n/\varepsilon))$ 迭代界降低到 $O(n^{3} \log(n/\varepsilon))$。此外，我们证明了关于近似帧缩放解大小的一个新界，涉及线性规划文献中研究的条件度量 $\barχ$，这可能具有独立的意义。
英文摘要：
The frame scaling problem is: given vectors $U := \{u_{1}, ..., u_{n} \} \subseteq \mathbb{R}^{d}$, marginals $c \in \mathbb{R}^{n}_{++}$, and precision $\varepsilon > 0$, find left and right scalings $L \in \mathbb{R}^{d \times d}, r \in \mathbb{R}^n$ such that $(v_1,\dots,v_n) := (Lu_1 r_1,\dots,Lu_nr_n)$ simultaneously satisfies $\sum_{i=1}^n v_i v_i^{\mathsf{T}} = I_d$ and $\|v_{j}\|_{2}^{2} = c_{j}, \forall j \in [n]$, up to error $\varepsilon$. This problem has appeared in a variety of fields throughout linear algebra and computer science. In this work, we give a strongly polynomial algorithm for frame scaling with $\log(1/\varepsilon)$ convergence. This answers a question of Diakonikolas, Tzamos and Kane (STOC 2023), who gave the first strongly polynomial randomized algorithm with poly$(1/\varepsilon)$ convergence for the special case $c = \frac{d}{n} 1_{n}$. Our algorithm is deterministic, applies for general $c \in \mathbb{R}^{n}_{++}$, and requires $O(n^{3} \log(n/\varepsilon))$ iterations as compared to $O(n^{5} d^{11}/\varepsilon^{5})$ iterations of DTK. By lifting the framework of Linial, Samorodnitsky and Wigderson (Combinatorica 2000) for matrix scaling to frames, we are able to simplify both the algorithm and analysis. Our main technical contribution is to generalize the potential analysis of LSW to the frame setting and compute an update step in strongly polynomial time that achieves geometric progress in each iteration. In fact, we can adapt our results to give an improved analysis of strongly polynomial matrix scaling, reducing the $O(n^{5} \log(n/\varepsilon))$ iteration bound of LSW to $O(n^{3} \log(n/\varepsilon))$. Additionally, we prove a novel bound on the size of approximate frame scaling solutions, involving the condition measure $\barχ$ studied in the linear programming literature, which may be of independent interest.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
