Spectral Analysis of Matrix Scaling and Operator Scaling

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.