Clustering a sparse matrix along diagonal line using row permutations only

Question

Given the binary pattern of a square sparse matrix. How can you move all non-zero elements towards diagonal line through row permutations. One possible cost function is sum the two norm of distances between each non zero element and the diagonal line.

Answer 1

This is more of an algorithms question, but one simple way would be to take the greedy approach. Keep evaluating which row swap would result in maximal improvement in your cost function. Repeat until the cost function stabilizes.