Exact Fractional Inference via Re-Parametrization Interpolation between Tree-Re-Weighted- and Belief Propagation- Algorithms
CoRR(2023)
摘要
Inference efforts – required to compute partition function, Z, of an Ising
model over a graph of N “spins" – are most likely exponential in N.
Efficient variational methods, such as Belief Propagation (BP) and Tree
Re-Weighted (TRW) algorithms, compute Z approximately minimizing respective
(BP- or TRW-) free energy. We generalize the variational scheme building a
λ-fractional-homotopy, Z^(λ), where λ=0 and
λ=1 correspond to TRW- and BP-approximations, respectively, and
Z^(λ) decreases with λ monotonically. Moreover, this
fractional scheme guarantees that in the attractive (ferromagnetic) case
Z^(TRW)≥ Z^(λ)≥ Z^(BP), and there exists a unique
(“exact") λ_* such that, Z=Z^(λ_*). Generalizing the
re-parametrization approach of and the loop
series approach of , we show how to express Z as a
product, ∀λ: Z=Z^(λ) Z^(λ), where the
multiplicative correction, Z^(λ), is an expectation over a
node-independent probability distribution built from node-wise fractional
marginals. Our theoretical analysis is complemented by extensive experiments
with models from Ising ensembles over planar and random graphs of medium- and
large- sizes. The empirical study yields a number of interesting observations,
such as (a) ability to estimate Z^(λ) with O(N^4) fractional
samples; (b) suppression of λ_* fluctuations with increase in N for
instances from a particular random Ising ensemble.
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