Strongly Polynomial Frame Scaling to High Precision

TL;DR

This work delivers a deterministic, strongly polynomial-time algorithm for frame scaling: given a frame $U$, marginals $c$, and precision $\varepsilon$, it computes a squared right scaling $z$ (with implicit left scaling) so that the leverage scores $\tau^U(z)$ match $c$ within $\varepsilon$ in $\ell_2$-norm, while preserving the isotropy condition. Building on the LSW framework, the authors use leverage-score based updates on a subset $T$ of columns and a Newton–Dinkelbach update to guarantee geometric progress in each iteration, achieving $O(n^3\log(n/\varepsilon))$ iterations and polynomial time per iteration. A key technical contribution is lifting the potential analysis to the frame setting and defining a proxy function $h_T^U$ that ties the progress to leverage-score gaps, enabling a strongly polynomial update mechanism. They also develop a regularization strategy tied to the $\bar{\chi}$ and $\rho$ condition measures to bound the bit complexity of scalings, and they show that the same approach yields improved iteration bounds for matrix scaling. Beyond the core algorithm, the paper connects to learning theory by applying the frame-scaling framework to halfspace learning and point-location problems, offering a path to robust, strongly polynomial learning methods with high-precision regularization.

Abstract

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.

Theorems & Definitions (83)

• Definition 1.1: Frame Scaling
• Theorem 1.2: BartheBLGelfandMatroid
• Theorem 1.3: Main Theorem (Informal)
• Definition 2.1
• Definition 2.3: Isotropic Frame Scaling Problem
• Theorem 2.4
• Definition 2.5
• Proposition 2.6
• proof
• Remark 2.7