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I'm interested in understanding whether a particular natural function on matrices, closely related to the permanent of a matrix, is invertible, and whether its inverse admits a simple closed form. The function $f$ I'm concerned with maps doubly stochastic $n \times n$ matrices to doubly stochastic $n \times n$ matrices, and is given as follows

$$ f_{ij}(A) = A_{ij} \frac{\operatorname{perm} \left( \tilde{A}_{ij} \right)}{\operatorname{perm} (A)} $$

where $\operatorname{perm}$ denotes the permanent function, and $\tilde{A}_{ij}$ denotes the $(i,j)$'th minor of $A$ — i.e.m the matrix gotten by removing the $i$-th row and $j$-th column of $A$. Observe that,

  • by the row [column] expansion formula for the permanent, $f(A)$ is doubly stochastic whenever $A$ is doubly stochastic.

  • $f$ is continuous, and each entry of $f(A)$ is the ratio of two multilinear functions of the entries of $A$.

I have three questions:

  1. Is $f$ invertible? If not, does $f$ at least have a right inverse, and is there such a right inverse that is continuous?

  2. If $f$ has an inverse, is there a simple closed form for $f^{-1}$? Otherwise, if $f$ has a continuous right-inverse, is there a simple closed form for this right inverse?

  3. If $f$ has an inverse (or continuous right-inverse), but the answer to (b) is no, can the inverse at least be (approximately) computed by a simple, preferably efficient (polynomial-time in n) procedure?

Rodrigo de Azevedo
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sd234
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    I'm interested to know where you came across this function $f$. I'm also interested to know whether you have calculated any examples, which might give you some idea of whether it is invertible, what an inverse might look like, etc. – Gerry Myerson May 03 '12 at 06:53
  • I don't think each entry of $f(A)$ is a multilinear function of the entries of $A$ -- the entries of $A$ also appear in the permanent in the denominator. – joriki May 03 '12 at 09:15
  • Joriki: You are right. I meant to say that the "interesting" part of $f$, the numerator, is a multilinear function. Will correct. – sd234 May 03 '12 at 12:24
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    Gerry: Where this function comes from, and why we want to invert it, is a long story, but here is a brief explanation. Consider the procedure that takes in a matrix $A$, and samples non-uniformly a random matching of the complete $n \times n$ bipartite graph, such that the probability of matching $M$ is proportional to the product of the weight of its edges as given by $A$. If you do the calculation, you observe that the random choice of $M$ contains $(i,j)$ with probability $f_{ij}(A)$. Now, why I'm interested in inverting $f$ would take much longer to explain... – sd234 May 03 '12 at 12:45
  • Thanks. So...have you calculated any examples? – Gerry Myerson May 04 '12 at 13:11
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    Gerry: The 2x2 case is simple. $f^-{1}(A)$ simply takes the square root of each entry of $A$, and rescales to restore stochasticity. The 3x3 case involves 9 variables, and appears nontrivial to invert, though doing so mechanically would be the next thing to try in order to see if the form of the inverse reveals some structure. – sd234 May 07 '12 at 03:38

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