Regression with embedded convolution

Question

I have a problem where the data I am getting is a convolution of the original data with some function and I am trying to solve the following equation for $A$ $$ Y = AX $$ where $Y \in \mathbb{R}^{n\times p}$, $X \in \mathbb{R}^{n\times p}$, $A \in \mathbb{R}^{p \times p}$, $Y$ and $X$ is the original unknown data and $A$ is unknown. Following is the data which I am receiving $$Y'(i) = h_i \circledast Y(i) \quad \forall i=1..p$$ $$X'(i) = h_i \circledast X(i) \quad \forall i=1..p$$ where $h_i$s are unknown, and $X'$ and $Y'$ are known. In short $X'$, $Y'$ are known, and $X$, $Y$, $A$ and $h_i$s are not known. I not able to find a way to solve for $A$. I would be really grateful if someone can give some pointers for it. I have asked the same question here https://stats.stackexchange.com/questions/348723