Spatial decay of correlations and efficient methods for computing partition functions.

David Gamarnik

Joint work with

Antar Bandyopadhyay (U of Chalmers), Dmitriy Rogozhnikov-Katz (MIT)

June, 2006

Talk Outline

• Partition functions. Where do we see them ?

• Computing partition functions. Monte Carlo method.

• Correlation decay.

• Our results: computation tree, correlation decay and • Deterministic algorithm for approximate computation of partition functions for matchings and colorings.• Structural results and large deviations.

• Conclusions

Partition functions - feature in

• statistical mechanics Gibbs measure and Ising models

• computer science and combinatorics counting problems

• queueing theory product form loss networks

• electrical engineeringcoding theory

• statistics bayesian networks

• Calls arrive as and request communication link

• Call is accepted only if no other link attached to is occupied

• Unaccepted call is lost

• Call duration is

Queueing Example: loss system with shared resources

• At any moment the set of occupied links is a matching

• The steady-state distribution is product form:

- partition function.

• Calls arrive as and occupy a node

• Call is accepted only if no neighbor is occupied

• Unaccepted call is lost

• Call duration is

Example II: multicasting in a communication network

• At any moment the set of occupied nodes is an independent set

• The steady-state distribution is product form:

- partition function.

• Calls arrive as and occupy a node and use frequency

• Call is accepted only if no neighbor is occupied and uses the same fr.

• Unaccepted call is lost

• Call duration is

Example III: multicasting with many frequencies

• At any moment the set of occupied nodes is a partial coloring

• The steady-state distribution is product form:

- partition function.

• Communication (matching) problem with

From queueing to statistical physics

- Gibbs distribution on Ising type models. Important object in stat mechanics.

- inverse temperature

- Monomer-dimer model.

• Matching problem with

From statistical physics to computer science

total number of matchings in the graph (counting)

Can we compute partition function?...

… easily when the underlying graph is a tree.

Example (independent sets)

This leads to

Theorem. Spitzer [75], Zachary [83,85], Kelly [85]. In -ary tree

Is independent from the boundary condition (correlation decay) if and only if

Ramanan, Sengupta, Zeidins, Mitra [2002] Related work on unicasting and multicasting on trees

Implication: if the graph is locally-tree like, then computing marginals is possible in the regime

Computing partition function in general

• Valiant [1979] -- #P complexity class. Exact counting is hard for most of the counting problems (matchings, independent sets, colorings, etc. )

• Focus – approximate counting.

Our contribution: - use of correlation decay for

- Deterministic (non-simulation based) algorithms for computing approximately partition functions for

• Matchings in low degree graphs

• Colorings in low degree graphs

- Structural properties of partition functions in special classes of graphs

Existing approaches for computing partition function

• Main approximation method: Markov Chain Monte Carlo (MCMC)

• The MCMC is based on

- computing the marginal distribution via simulation.

- reducing partition function to marginals (cavity method).Jerrum, Valiant & Vazirani [86]

• Technical challenge: establishing rapid mixing

Computing partition functions using MCMC

Jerrum [95]. Coloring

Vigoda [2000]. Coloring Coloring

Jerrum & Sinclair [89] Matchings

Dyer, Frieze & Kannan [91] Volume of a convex body.

Jerrum, Sinclair & Vigoda [2004]. Permanents

(Temporal) Decay of correlations in Markov chains

A Markov chain with transition matrix satisfies decay of correlation (mixes)

if and only if it is aperiodic

(Spatial) Decay of correlations

Same thing, but time is replaced by a “spatial” distance

Correlation DecayA sequence of spatially (graph) related random variables exhibits a decay of correlation (long-range independence),if when is large

Principle motivation - statistical phyisics. Uniqueness of Gibbs measures on infinite lattices, Dobrushin [60s].

What is known about correlation decay ?

Spitzer [75], Zachary [83,85], Kelly [85]. Independent sets -ary tree

J. van den Berg [98] Matchings

Goldberg, Martin & Paterson [05]. Coloring. General graphs

Jonasson [01]. Coloring. Regular trees

Link between Correlation Decay and rapid mixing of MC:

• CD implies rapid mixing in subexp. growing graphs. • Converse not true Kenyon, Mossel & Peres [01].

Our results:

Theorem I. There exists a deterministic algorithm for computing approximately the partition function corresponding to matchings in graphs with constant degree (deterministic FPTAS), for arbitrary

Related work: Weitz [2005]. Self-avoiding walk based algorithm for counting independent sets when

Algorithm and proof:

Step I. Reduce computing partition function to computing marginals (cavity method)

Thus computing marginals implies computing the partition function

Step II. Cavity recursion

Step II. Cavity recursion

Step II. Cavity recursion

Algorithm: repeat the recursion times.

Initialize at the bottom arbitrarily.

Compute recursively.

- Computation tree

Proposition. The computation tree satisfies the decay of correlation property

Proof: look at the recursion function:

Introduce change of variables:

Mean Value Theorem:

- contraction

Theorem II. There exists a deterministic algorithm for computing approximately the number of list colorings in triangle-free graphs when the size of each list is constant and

for all nodes

Cavity recursion

Cavity recursion

x

x

x

Cavity recursion

We establish correlation decay for this recursion

x

x

x

Why can’t we use conventional decay of correlation directly for counting by computingmarginals locally for small (constant) ?

Problem:

We need accuracy in order to have accuracy

But:

Theorem III. The partition function of independent sets in every r-regular locally tree-like graphs satisfies

when

Structural results

The decay of correlation property implies the following large deviations results:

Queueing/large deviations interpretation

1. In a multicasting model (independent sets) the probability that nobody is transmitting a signal is

2. The probability that the set of active nodes is is given as

These results are not “provable” using MCMC technique

Structural results

Theorem IV. The partition function of the number of q-colorings in every r-regular graph with large girth satisfies

Note: removing a node when computing marginals destroys regularity

A fix comes from a rewiring trick Mezard-Parisi [05].

Lemma. The rewiring operation can be performed on pairs of nodes without creating small cycles.

Final thoughts and goals

• Queueing and stationarity.

Consider a queueing version of the “matching” problem. Assume FIFO.

Does the loss of stationarity occur before or after onset of long-range dependence?

Final thoughts and goals

• Create an implementable version of our algorithm (aka Belief Propagation). Our algorithm is only nominally efficient.

• Combining algorithm with importance sampling to handle large degree instances.

• Other counting problems: permanent, volume of a polyhedron.

• What other structures have the underlying computation tree satisfy the correlation decay property? Markov random fields?