This is about the matrix balancing problem.
We have three sets of data:
- A matrix with with entries \(\color{darkblue}A^0_{i,j}\ge 0\).
- Row- and column-totals \(\color{darkblue}r_i\) and \(\color{darkblue}c_j\).
The \(\color{darkblue}A^0\) matrix is collected from different sources than the row- and column-totals. So the matrix elements don't sum up to our totals. The problem is finding a nearby matrix \(\color{darkred}A\), so the row and column totals are obeyed. In addition, we want to preserve the sparsity pattern of \(\color{darkblue}A^0\): zeros should stay zero. And also: we don't want to introduce negative numbers (preserve the signs). More formally:
| Matrix Balancing Problem |
|---|
| \[\begin{align}\min\>&{\bf{dist}}(\color{darkred}A,\color{darkblue}A^0)\\ & \sum_i \color{darkred}A_{i,j} = \color{darkblue}c_j && \forall j\\ & \sum_j \color{darkred}A_{i,j} = \color{darkblue}r_i && \forall i \\&\color{darkred}A_{i,j}=0 &&\forall i,j|\color{darkblue}A^0_{i,j}=0\\ &\color{darkred}A_{i,j}\ge 0 \end{align} \] |
 |
| Approximate the matrix subject to row- and column-sum constraints |
