Factorisation of stochastic matrices

Here we derive updates rules for the approximation of a row stochastic matrix by the product of two lower-rank row stochastic matrices using gradient descent. Such a factorisation corresponds to a decomposition

$$p(n|m)=\sum _{k}p(n|k)\cdot p(k|m)$$

Both the sum of squares and row-wise cross-entropy functions are considered.