Contents
--------
1. Background
2. Provided code
3. Known rough edges
4. BibTeX entries for the references
1. Background
-------------
The algorithm of Swendsen and Wang (1987) provides amazingly fast updates on
classical Potts/Ising models on a grid. The original paper mentions some
generalizations, although the exposition of Edwards and Sokal (1988) makes it
very clear how to derive Swendsen-Wang like algorithms for many models.
Although the Swendsen-Wang algorithm (S-W) is quite broadly applicable in
theory, it won't work well on some distributions. The oft-cited papers for this
observation are by Gore and Jerrum (1997, 1999). These works describe problems
with a 1st order phase transition which would also break Gibbs sampling and any
standard MCMC method. It is also fairly well known that "frustrated" systems can
break S-W, even in some cases where Gibbs sampling works fine. I have seen
Boltzmann Machine models fitted to images that won't move at all with S-W but
mix fine with Gibbs sampling. I've also seen it work much better than Gibbs
sampling (Stern et al., 2004). S-W is a really neat algorithm, but you have to
work out if your problem is appropriate.
2. Provided code
----------------
This directory contains code for S-W sampling on fully-connected undirected
graphs of binary variables with pairwise potentials and site biases. This class
of models includes Ising models and Boltzmann machines. The code is for an Ising
model parameterization, {+1,-1} variables, but there is a wrapper for a
Boltzmann machine parameterization involving {0,1} variables. The code supports
non-fully connected graphs by setting weights to zero, it could be more
efficient by dealing with sparse graphs properly however.
The most expensive part of a S-W is (or should be) the percolation step. I used
an algorithm by Newman and Ziff (2001). If the graph has special structure, then
better algorithms may apply. For example there is a fast percolation method that
has been used for S-W on planar graphs (Sweeny, 1983).
Be very careful that the parameters (especially the weights matrix) that you are
passing to the provided functions specify the distribution that you think they
do. There are many different conventions for the parameters. I have tried to
explain the conventions that I am using in the functions, but check on a tiny
problem that we agree.
The tests directory contains some very simple sanity checks. test_distribution.m
should also help confirm what distributions my codes claim to sample from as it
explicitly constructs the distributions for a tiny test problem to compare
against. The tests directory also demonstrates how to estimate marginals via a
so-called Rao-Blackwellization using conditional probabilities from the FKSW
joint distribution identified by Edwards and Sokal.
The orig directory contains an old, very first stab at Matlab code for
Swendsen-Wang after I had just worked through the details for the first time.
3. Known rough edges
--------------------
I should use Matlab's memory management functions in the mex file so I don't
risk leaking memory if the code is interrupted.
I shouldn't use the standard C library random number generator in the mex file.
If you're serious about using this code, get around to replacing that line of
code with a call to a decent generator.
I haven't profiled the code, or worked on performance. Real problems won't have
fully connected graphs and the code should exploit that.
Look for better codes and let me know of URLs I should list here. There will be
statistical physicists or vision people out there with better code.
4. BibTeX entries for the references
------------------------------------
@article{swendsen1987,
title={Nonuniversal critical dynamics in {M}onte {C}arlo simulations},
author={Swendsen, R. H. and Wang, J. S.},
journal={Physical Review Letters},
volume={58},
number={2},
pages={86--88},
year={1987}
}
@article{edwards1988,
title={Generalizations of the {F}ortuin-{K}asteleyn-{S}wendsen-{W}ang representation and {M}onte {C}arlo algorithm},
author={Robert G. Edwards and A. D. Sokal},
journal={Physical Review},
volume={38},
pages={2009--2012},
year={1988}
}
@inproceedings{gore1997,
title={The {S}wendsen-{W}ang process does not always mix rapidly},
author={Gore, V. K. and Jerrum, M. R.},
booktitle={Proceedings of the twenty-ninth annual ACM symposium on Theory of computing},
pages={674--681},
year={1997},
publisher={ACM Press New York, NY, USA}
}
@article{gore1999,
title={The {S}wendsen--{W}ang process does not always mix rapidly},
author={Gore, V. K. and Jerrum, M. R.},
journal={Journal of Statistical Physics},
volume={97},
number={1},
pages={67--86},
year={1999}
}
@inproceedings{stern2004,
title={Modelling uncertainty in the game of go},
author={David Stern and Thore Graepel and David J. C. MacKay},
booktitle={Advances in Neural Information Processing Systems 17: Proceedings of the 2004 Conference},
editor={Lawrence K. Saul and Yair Weiss and L{\'e}on Bottou},
year={2005},
publisher={MIT Press}
}
@article{newman2001,
title={Fast {M}onte {C}arlo algorithm for site or bond percolation},
author={M. E. J. Newman and R. M. Ziff},
journal={Physical Review E},
volume={64},
number={016706},
month={June},
year={2001},
url={http://prola.aps.org/abstract/PRE/v64/i1/e016706}
}
@article{sweeny1983,
title={{M}onte {C}arlo study of weighted percolation clusters relevant to the {P}otts models},
author={Mark Sweeny},
journal={Physical Review B},
volume={27},
number={7},
pages={4445--4455},
year={1983},
url={http://prola.aps.org/abstract/PRB/v27/i7/p4445_1}
}