Strongly Polynomial Frame Scaling to High Precision

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.