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 $\bar{\chi}$ studied in the linear programming literature, which may be of independent interest.

翻译：框架缩放问题定义为：给定向量 $U := \{u_{1}, ..., u_{n} \} \subseteq \mathbb{R}^{d}$、边际 $c \in \mathbb{R}^{n}_{++}$ 及精度 $\varepsilon > 0$，寻找左缩放 $L \in \mathbb{R}^{d \times d}$ 与右缩放 $r \in \mathbb{R}^n$，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 $\|v_{j}\|_{2}^{2} = c_{j}, \forall j \in [n]$，且误差在 $\varepsilon$ 以内。该问题广泛出现在线性代数与计算机科学的多个领域。本文提出一种具有 $\log(1/\varepsilon)$ 收敛速度的强多项式框架缩放算法。这回答了Diakonikolas、Tzamos 与 Kane（STOC 2023）提出的问题：他们在特殊情形 $c = \frac{d}{n} 1_{n}$ 下给出了首个具有 poly$(1/\varepsilon)$ 收敛速度的强多项式随机算法。我们的算法是确定性的，适用于一般 $c \in \mathbb{R}^{n}_{++}$，且仅需 $O(n^{3} \log(n/\varepsilon))$ 次迭代，而 DTK 算法需要 $O(n^{5} d^{11}/\varepsilon^{5})$ 次迭代。通过将 Linial、Samorodnitsky 与 Wigderson（Combinatorica 2000）的矩阵缩放框架推广至框架情形，我们得以简化算法与分析。本文的主要技术贡献在于：将 LSW 的势函数分析推广到框架设定，并在强多项式时间内计算更新步长，从而实现每次迭代的几何进展。事实上，我们可调整所得结果以给出强多项式矩阵缩放的改进分析，将 LSW 的 $O(n^{5} \log(n/\varepsilon))$ 迭代界降至 $O(n^{3} \log(n/\varepsilon))$。此外，我们证明了一个关于近似框架缩放解规模的新颖上界，其中涉及线性规划文献中研究的条件度量 $\bar{\chi}$，该结果可能具有独立意义。