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| Format: | Preprint |
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2023
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| Online-Zugang: | https://arxiv.org/abs/2301.10369 |
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| _version_ | 1866916478704418816 |
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| author | Behjoo, Hamidreza Chertkov, Michael |
| author_facet | Behjoo, Hamidreza Chertkov, Michael |
| contents | Computing the partition function, $Z$, of an Ising model over a graph of $N$ \enquote{spins} is most likely exponential in $N$. Efficient variational methods, such as Belief Propagation (BP) and Tree Re-Weighted (TRW) algorithms, compute $Z$ approximately by minimizing the respective (BP- or TRW-) free energy. We generalize the variational scheme by building a $λ$-fractional interpolation, $Z^{(λ)}$, where $λ=0$ and $λ=1$ correspond to TRW- and BP-approximations, respectively. This fractional scheme -- coined Fractional Belief Propagation (FBP) -- guarantees that in the attractive (ferromagnetic) case $Z^{(TRW)} \geq Z^{(λ)} \geq Z^{(BP)}$, and there exists a unique (\enquote{exact}) $λ_*$ such that $Z=Z^{(λ_*)}$. Generalizing the re-parametrization approach of \citep{wainwright_tree-based_2002} and the loop series approach of \citep{chertkov_loop_2006}, we show how to express $Z$ as a product, $\forall λ:\ Z=Z^{(λ)}{\tilde Z}^{(λ)}$, where the multiplicative correction, ${\tilde Z}^{(λ)}$, 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. Our empirical study yields a number of interesting observations, such as the ability to estimate ${\tilde Z}^{(λ)}$ with $O(N^{2::4})$ fractional samples and suppression of variation in $λ_*$ estimates with an increase in $N$ for instances from a particular random Ising ensemble, where $[2::4]$ indicates a range from $2$ to $4$. We also discuss the applicability of this approach to the problem of image de-noising. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2301_10369 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Exact Fractional Inference via Re-Parametrization & Interpolation between Tree-Re-Weighted- and Belief Propagation- Algorithms Behjoo, Hamidreza Chertkov, Michael Machine Learning Statistical Mechanics Computing the partition function, $Z$, of an Ising model over a graph of $N$ \enquote{spins} is most likely exponential in $N$. Efficient variational methods, such as Belief Propagation (BP) and Tree Re-Weighted (TRW) algorithms, compute $Z$ approximately by minimizing the respective (BP- or TRW-) free energy. We generalize the variational scheme by building a $λ$-fractional interpolation, $Z^{(λ)}$, where $λ=0$ and $λ=1$ correspond to TRW- and BP-approximations, respectively. This fractional scheme -- coined Fractional Belief Propagation (FBP) -- guarantees that in the attractive (ferromagnetic) case $Z^{(TRW)} \geq Z^{(λ)} \geq Z^{(BP)}$, and there exists a unique (\enquote{exact}) $λ_*$ such that $Z=Z^{(λ_*)}$. Generalizing the re-parametrization approach of \citep{wainwright_tree-based_2002} and the loop series approach of \citep{chertkov_loop_2006}, we show how to express $Z$ as a product, $\forall λ:\ Z=Z^{(λ)}{\tilde Z}^{(λ)}$, where the multiplicative correction, ${\tilde Z}^{(λ)}$, 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. Our empirical study yields a number of interesting observations, such as the ability to estimate ${\tilde Z}^{(λ)}$ with $O(N^{2::4})$ fractional samples and suppression of variation in $λ_*$ estimates with an increase in $N$ for instances from a particular random Ising ensemble, where $[2::4]$ indicates a range from $2$ to $4$. We also discuss the applicability of this approach to the problem of image de-noising. |
| title | Exact Fractional Inference via Re-Parametrization & Interpolation between Tree-Re-Weighted- and Belief Propagation- Algorithms |
| topic | Machine Learning Statistical Mechanics |
| url | https://arxiv.org/abs/2301.10369 |