SIAM Journal on Computing, Volume 50, Issue 3, Page 1034-1102, January 2021.
We present a spectral analysis of a continuous scaling algorithm for matrix scaling and operator scaling. The main result is that if the input matrix or operator has a spectral gap, then a natural gradient flow has linear convergence. This implies that a simple gradient descent algorithm also has linear convergence under the same assumption. The spectral gap condition for operator scaling is closely related to the notion of quantum expander studied in quantum information theory. The spectral analysis also provides bounds on some important quantities of the scaling problems, such as the condition number of the scaling solution and the capacity of the matrix and operator. These results can be used in various applications of scaling problems, including matrix scaling on expander graphs, permanent lower bounds on random matrices, the Paulsen problem on random frames, and Brascamp--Lieb constants on random operators. In some applications, the inputs of interest satisfy the spectral condition and we prove significantly stronger bounds than the worst case bounds.

中文翻译：

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矩阵缩放和算子缩放的频谱分析

SIAM Journal on Computing，第 50 卷，第 3 期，第 1034-1102 页，2021 年 1 月。
我们提出了矩阵缩放和算子缩放的连续缩放算法的频谱分析。主要结果是，如果输入矩阵或算子存在谱间隙，则自然梯度流具有线性收敛性。这意味着在相同的假设下，简单的梯度下降算法也具有线性收敛性。算子缩放的光谱间隙条件与量子信息理论中研究的量子扩展器的概念密切相关。谱分析还提供了缩放问题的一些重要数量的界限，例如缩放解的条件数以及矩阵和算子的容量。这些结果可用于缩放问题的各种应用，包括扩展图上的矩阵缩放、随机矩阵的永久下界、随机框架上的保尔森问题，以及随机算子上的 Brascamp--李布常数。在某些应用中，感兴趣的输入满足频谱条件，我们证明了比最坏情况下的边界强得多的边界。