Spectral Analysis of Matrix Scaling and Operator Scaling

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