I have $q$ $n$-dimensional vectors $\vec y_i$ and a matrix $\hat B$ of shape $n\times m$. I'm looking for $q$ $m$-dimensional vectors $\vec x_i$ such that:
- $\vec y_i=\hat B \vec x_i$
- each vector $\vec x_i$ represents probability distribution (each entry is non-negative and the $L_1$ norm is 1)
Associated issues are that:
- vectors $\vec y_i$ are very noisy
- matrix $\hat B$ can be estimated with bigger accuracy, but it may be rather sparse
How does one approach such problem?