On the inversion of the Valiant function of the rank rigidity of a matrix

The rank function $\mathrm{rank}(A,k)$ of a matrix $A$ is the minimal rank of a matrix obtained from $A$ by changing no more than $k$ of its entries. For an arbitrary matrix, we obtain an upper boundary of $\mathrm{rank}(A,k)$. For rigid matrices, we establish a smooth lower boundary and a precise formula for $\mathrm{rank}(A,k)$. Alos, we show that the rank function of a rigid matrix inverses its regidity function. For rigid matrices, an interpretation of the inverse function of the rigidity function is given.