MATLAB: Row & Column Wise Normalisation

column-wisedoubly stochastic matrixMATLABnormalisenormalise columnsnormalise rowsrow-wisesinkhornsinkhorn-knopp

Objective: Normalise a matrix such that all rows and columns sum to 1.
The below normalises each column, then row and repeats until row and column totals, equal one another.
This seems to work for randomly generated arrays.
However, the data I wish to use it on has some zeros – and that is generating lots of NaN and Infs, which is making things quite messy and sometimes when running the while loop won't execute (no error message, it just hops over it)
I've tried changing the while condition to be rounded to 3 decimal places (because that's good enough) but still no success.
a = rand(7)
rows = sum(a,2) % orginal row totals
cols = sum(a,1) % original col totals
b = a;
i = 1; % for counting how many iterations
while sum(b,1,"omitnan") ~= sum(b,2,"omitnan")' %when column totals == row totals, stop.
    b = b ./ sum(b,1,"omitnan"); %divide by col totals
    b = b ./ sum(b,2,"omitnan"); %divide by row totals
    i = i + 1;
end
i %how many loops
b % normalised output
brows = sum(b,2,"omitnan") %check that all rows sum 1
bcols = sum(b,1,"omitnan") %check that all cols sum 1.
attached are two 7 x 7 matrices. These are the desired input for a.
Suggestions welcome.
edit:
The margfit function (row 345 – 376) in link below, is (I think) what I am trying to implement. My python is non-existant
https://github.com/GoricaB/Land-cover-validation/blob/master/pts_lcval.py

Best Answer

For a non-negative square matrix, the attached article mentions necessary and sufficient conditions (p. 3, Theorem 1) both for the normalization you are trying to achieve to be possible and for the alternating row/column normalization approach (the Sinkhorn-Knopp algorithm ) to work. The required condition for both are the same. So basically, if you are seeing Infs and NaNs in your iterations, the normalization is known to be impossible from the get-go.
The condition is:
"A necessary and sufficient condition ... is that A has total support"
The given matrix A having total support means that for every non-zero element A(i,j)>0, a column permutation Ap of A exists such that Ap has only strictly positive elements on the diagonal, one of which is A(i,j).
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