Sinkhorn-Knopp algorithm for matrix normalisation

The Sinkhorn-Knopp algorithm takes a matrix A and finds diagonal matrices D and E such that if M = DAE the sum of each column and each row of M is unity. The method is, in effect, to alternately normalise the rows and the columns of the matrix. This function is an efficient implementation which does not actually carry out normalisation until the iteration is complete and which does not use the transpose of A.

A must be non-negative. If there are zeros in A the algorithm may not converge, depending on their distribution. A maximum number of iterations and/or an error tolerance may be set if required.

A matrix whose rows and columns all sum to unity is termed "doubly stochastic". Such matrices have various applications, including web page ranking.

Reference:

Philip A. Knight (2008) The Sinkhorn–Knopp Algorithm: Convergence and Applications. SIAM Journal on Matrix Analysis and Applications 30(1), 261-275. doi: 10.1137/060659624