The Birkhoff polytope is a convex polytope whose points are doubly stochastic matrices and whose vertices are permutation matrices.
Questions tagged [birkhoff-polytopes]
36 questions
29
votes
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Does Birkhoff - von Neumann imply any of the fundamental theorems in combinatorics?
I recently had the occasion to think about Hall's Marriage Theorem for the first time since my undergraduate combinatorics class more than a decade ago. Reading the wikipedia article linked above, I was interested to see that it is regarded as…
Pete L. Clark
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How do I generate doubly-stochastic matrices uniform randomly?
A doubly-stochastic matrix is an $n \times n$ matrix $P$ such that
$$ \sum_{i=1}^n p_{ij} = \sum_{j=1}^n p_{ij} = 1 $$
where $p_{ij}\ge 0$. Can someone please suggest an algorithm for generating these matrices uniform randomly?
Henry B.
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What's the algorithm of finding the convex combination of permutation matrices for a doubly stochastic matrix?
According to Birkhoff, $n$-by-$n$ stochastic matrices form a convex polytope whose extreme points are precisely the permutation matrices. It implies that any doubly stochastic matrix can be written as a convex combination of finitely many…
xzhu
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11
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Proof that the set of doubly-stochastic matrices forms a convex polytope?
Does the set of all doubly-stochastic matrices form a convex polytope? In general, I wonder how the proofs of convexity and geometry can be established for sets of matrices of this kind? Anything to do with the birkhoff's theorem!? Am not sure, if…
qlinck
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8
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$n$-sphere enclosing the Birkhoff polytope
I am not a mathematician by training, so please feel free to correct my logic or descriptions when necessary:
Let $P$ denote a $\textit{permutation matrix}$:
$$
\begin{equation}
P := \{X \in \{0,1\}^{n\times n} : X \mathbf{1}_n =…
Tolga Birdal
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7
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Projection onto Birkhoff Polytope
Suppose we would like to compute the Euclidean projection of an arbitrary matrix $A$ onto the Birkhoff polytope, the set of doubly-stochastic matrices.
Under some conditions on $A$, Sinkhorn's algorithm returns two diagonal matrices $D_1,D_2$ such…
cong
- 173
7
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1 answer
How wide is the Birkhoff Polytope?
Now also posted on Math Overflow.
Define the width of a polytope $P \subset \mathbb R^d$ as the minimum length of the interval $\{v \cdot p:p \in P\}$ for $v$ in the unit sphere. In other words the width is the smallest number $W$ such that you can…
Daron
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Birkhoff representation of a stochastic matrix
From the Birkhoff theorem, it is known that every doubly stochastic matrix can be written as a convex combination of permutation matrices, although this representation might not be unique.
Assume that a stochastic matrix is given. How can I find a…
5
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1 answer
What are the facets of the Birkhoff Polytope when $n=2$?
I've read in several sources that the number of facets of the Birkhoff polytope $\mathcal{B}(n)$ is $n^2$.
Is this supposed to hold when $n=2$? Since $\mathcal{B}(2)$ has dimension $1$, the facets would be the two $0$-dimensional vertices, which…
M47145
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4
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The permutation matrices are the doubly stochastic matrices with the highest Frobenius norm
In a 2013 talk, Alexandre d'Aspremont did claim the following:
Among all doubly stochastic matrices, the rotations, hence, the permutation matrices, have the highest Frobenius norm
I had never encountered this result. Searching for it on…
Rodrigo de Azevedo
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3
votes
1 answer
Birkhoff-von Neumann Theorem
I am reading from Linear Algebra in Action by Dym and am working through the proof of the Birkhoff-von Neumann Theorem. For the sake of clarity, ill write up everything until the point where I get lost.
Theorem: Let $P \in \mathbb{R}^{n\times n}$ be…
Alex Sampson
- 103
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0 answers
Can Sinkhorn's algorithm reach any point on the Birkhoff polytope?
It is well known that Sinkhorn's algorithm converges to a doubly stochastic matrix given a non-negative square input matrix.
Knowing that Sinkhorn's algorithm always produces a DSM, I am interested in the opposite: Can every possible DSM be produced…
dopexxx
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Finding all integer solutions for a Linear Program over the Birkhoff Polytope
I have the situation where the Birkhoff Polytope is the space of all valid solutions to a linear function I am interested in maximizing. It is my understanding that, because the vertices of the Birkhoff polytope are the permutation matrices,…
2
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1 answer
Prove that a matrix can be written as a sum of permutation matrices
Given a square matrix $A$ of size $n$ whose entries are non-negative integers and where the sum of each column and row is equal to $k$, prove that $A$ can be written as a sum of $k$ permutation matrices.
First, it is obvious to see that a sum o…
314canelas
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What is the linear description of a transformation of Birkhoff polytope?
Let $I, J$ be finite sets and $|I|=|J|=n$, Let $F$ be a Birkhoff polytope formed by the convex hull of $n\times n$ doubly stochastic matrices:
$$F=\{R^{I\times J}_+: \sum_j x( i,j)=1,\forall i\in I,
\sum_i x( i,j)=1,\forall j\in J
\}$$
Suppose …
sam
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