A matrix $A \in \mathbb{R}^{m \times n}$ is a stochastic matrix if $A \in [0, 1]^{m \times n}$ and $\sum_j A_{ij} = 1, \forall i \in [n]$. Let $\mathcal{S_{m, n}}$ be the set of $m \times n$ stochastic matrices.
If we have an optimization problem on $S_{m, n}$: $$\arg \min_{A \in \mathcal{S_{m,n }}} f(A)$$ with $f: \mathbb{R}^{m \times n} \to \mathbb{R}$, how can we solve it via, e.g., gradient descent?
A possible way is to use any existing optimization algorithm on $\mathbb{R}^{m \times n}$ with a softmax transform on each row to find $\arg \min_{A \in \mathbb{R}^{m \times n}} f(\operatorname{row-wise-softmax}(A))$.
For doubly stochastic matrices, we have existing methods where the set of all doubly stochastic matrices forms a manifold and we may solve the optimization problem on the manifold.
[1] https://www.manopt.org/manifold_documentation_multinomialdoublystochastic.html
[2] https://arxiv.org/abs/1802.02628
Can we have a manifold for stochastic matrices too? Or can we have other methods to solve optimization problems on stochastic matrices?