I have a vector $\mathbf{x}$ of size $m\cdot n$ of zeros and ones, i.e., $\mathbf{x}\in\{0,1\}^{m\cdot n}$ and a matrix $\mathbf{A}$ of size $\left(m\cdot n+m+n+1\right)\times\left(m\cdot n\right)$ of reals, i.e., $\mathbf{A}\in\mathbb{R}^{\left(m\cdot n+m+n+1\right)\times\left(m\cdot n\right)}$.
The vector $\mathbf{x}$ has $O(m\cdot n)$ zeros and the matrix $\mathbf{A}$ has $O(m^2\cdot n+n^2\cdot m)$ zeros.
I would like to know the complexity of multiplying $\mathbf{A}$ by $\mathbf{x}$. Using a naive analysis, the complexity would be of the order of $O(m^2\cdot n^2)$ but I think that we can give a more rigorous analysis given that $\mathbf{x}$ and $\mathbf{A}$ have a lot of zeros elements.
Can you please give me some ideas, references?