http://www.yaroslavvb.com/research/qr/doubly-stochastic/doubly-stochastic.html

As Jeremy pointed out, Birkhoff's theorem provides necessary and sufficient conditions for a matrix to be doubly stochastic -- it must be a convex combination of permutation matrices. It seems hard to sample from the set of valid convex combinations though. For instance, here's a set of 3 dimensional slices through the set of valid convex combinations for 3x3 matrices (there are 6 permutation matrices, so the space is 5 dimensional).

Update: here's the updated notebook with 2 more algorithms, suggested by Dr.Teh and Jeremy below. Turns out that sampling random convex combinations is fairly easy -- almost all the time uniformly sampled convex combination will produce a doubly-stochastic matrix. However, matrices end up looking pretty uniform

http://www.yaroslavvb.com/research/qr/doubly-stochastic/doubly-stochastic3.html

## 11 comments:

Your second approach reminds me of the following theorem (taken from Combinatorial Matrix Theory by Brualdi and Ryser):

A matrix A is doubly stochastic iff there exists permutation matrices P_1, ..., P_t and real numbers c_1, ... , c_t such that

A = c_1 P_1 + ... + c_t P_t

and

c_1 + ... + c_t = 1.

I have a question: why won't iteratively normalizing columns and rows work? Isn't it IPF? cheers, -yw

Jeremy -- thanks for the pointer. I also found it as Thm. 8.7.1 in Horn and Johnson

Dr Teh -- you are right, it's IPF, I made a mistake of running it for too few iterations, if I run it longer it tends to converge (within 10^-10 of doubly stochastic) after 30 iterations for 3x3 matrices, less for larger matrices

http://www.yaroslavvb.com/research/qr/doubly-stochastic/doubly-stochastic2.html

In fact, convergence of IPF to doubly-stochastic matrix was proven back in 1964

(convergence is guaranteed only if all matrix entries are strictly positive)

Isn't this simpler and equivalent? In Matlab notation: [T,~]=qr(rand(n));T=T.^2;

Excellent machine learning blog,thanks for sharing...

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