How would you encourage an orthogonal matrix to be a permutation matrix?

Asked Dec 23 '21 at 02:35

Active May 07 '23 at 22:12

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We want to solve an optimization problem over the space of permutation matrices (square binary matrices with exactly one entry of $1$ in each row and each column and $0$s elsewhere) with size $n \times n$, and we relax the constraint so that we operate optimization for orthogonal matrices ($P^TP=PP^T=I$), how would you encourage the matrices to be permutation matrices?

I have tried to add regularization on $\sum_{i,j} P_{ij}^4$. Would you recommend something else?

edited May 07 '23 at 22:12

Rodrigo de Azevedo

23,223

asked Dec 23 '21 at 02:35

Vezen BU

2,320

You can formulate it as an integer linear program. If each row/column sums to 1 and each entry is in the interval $[0,1]$, and all entries are integers, then it is a permutation matrix. – Jair Taylor Dec 23 '21 at 02:49
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"how would you encourage the matrices to be permutation matrices?" Talk to them in private. Never scream at an orthogonal matrix. – markvs Dec 23 '21 at 03:02
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An orthogonal matrix whose entries are all $\ge 0$ is a permutation matrix. – Robert Israel Dec 23 '21 at 03:49
@JairTaylor Thank you for your comment! In the case you mentioned, we are approximating permutation matrices by doubly stochastic matrices. But I want to try the alternative from the direction of orthogonal matrices. – Vezen BU Dec 23 '21 at 04:25
@RobertIsrael Thanks for your comment! Do you mean adding a penalty on negative entries? – Vezen BU Dec 23 '21 at 04:26
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That would do it. – Robert Israel Dec 23 '21 at 06:09
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Related – Rodrigo de Azevedo May 07 '23 at 22:15
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If you have 1 minute, you might want to hear Alexandre d'Aspremont about his "tricks". – Rodrigo de Azevedo May 11 '23 at 19:11
Thanks! @RodrigodeAzevedo – Vezen BU May 11 '23 at 22:24

How would you encourage an orthogonal matrix to be a permutation matrix?

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