Arbitrary sequences of single row or column normalizations converge to a doubly stochastic matrix that depends only on the positive supported part of the initial matrix, enabling a decentralized random walk algorithm with uniform stationary distribution.
On Schrodinger's bridge problem
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In the first part of this paper we generalize the result of Georgiou-Pavon that a positive square matrix can be scaled uniquely to a column stochastic matrix which maps a given positive probability vector to another given positive probability vector. In the second part of this paper we prove that a positive quantum channel can be scaled to another positive quantum channel which maps a given positive definite density matrix to another given positive definite density matrix using Brower's fixed point theorem. This result proves the Georgiou-Pavon conjecture for two positive definite density matrices in their recent paper. We show uniqueness of fixed points for certain two positive definite density matrices.
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From Local Updates to Global Balance: A Framework for Distributed Matrix Scaling
Arbitrary sequences of single row or column normalizations converge to a doubly stochastic matrix that depends only on the positive supported part of the initial matrix, enabling a decentralized random walk algorithm with uniform stationary distribution.