Region Graph Partition Function Expansion and Approximate Free Energy Landscapes: Theory and Some Numerical Results

Abstract

Graphical models for finite-dimensional spin glasses and real-world combinatorial optimization and satisfaction problems usually have an abundant number of short loops. The cluster variation method and its extension, the region graph method, are theoretical approaches for treating the complicated short-loop-induced local correlations. For graphical models represented by non-redundant or redundant region graphs, approximate free energy landscapes are constructed in this paper through the mathematical framework of region graph partition function expansion. Several free energy functionals are obtained, each of which use a set of probability distribution functions or functionals as order parameters. These probability distribution function/functionals are required to satisfy the region graph belief-propagation equation or the region graph survey-propagation equation to ensure vanishing correction contributions of region subgraphs with dangling edges. As a simple application of the general theory, we perform region graph belief-propagation simulations on the square-lattice ferromagnetic Ising model and the Edwards-Anderson model. Considerable improvements over the conventional Bethe-Peierls approximation are achieved. Collective domains of different sizes in the disordered and frustrated square lattice are identified by the message-passing procedure. Such collective domains and the frustrations among them are responsible for the low-temperature glass-like dynamical behaviors of the system.

Appendices

Appendix A: Derivation of Parent-to-Child Message-Passing Equation

Here we give a detailed derivation of Eqs. (52) and (53). These equations are valid for a non-redundant region graph R. We assume R to be non-redundant in this whole section.

First, the region subgraph formed by region γ and all its descendants is a connected tree (see the blue-shaded area of Fig. 5). This property ensures the equivalence of (51) with (49). Notice that a region μ∈B _γ may point to two or more regions of the set I _γ .

Applying the definition (24) and then the two identities (21) and (22), we obtain that

(76)

The set R _b ∖I _γ contains regions of subgraph R _b except those also belonging to set I _γ , and similarly for R _j ∖I _γ .

Because R _b and R _j are two connected tree subgraphs, we have

$$ \everymath{\displaystyle} \begin{array}{l} \sum _{\eta \in R_b \backslash I_\gamma} c_\eta = \sum _{\nu\in R_b \cap I_\gamma} \sum _{\{(\mu\rightarrow \nu)| \mu\in B_\gamma\}} \sum _{\alpha \in R_b^{\mu\rightarrow \nu}} c_\alpha ,\\[12pt] \sum _{\eta \in R_j \backslash I_\gamma} c_\eta = \sum _{\nu\in R_j \cap I_\gamma} \sum _{\{(\mu\rightarrow \nu) | \mu \in B_\gamma\}} \sum _{\alpha \in R_j^{\mu\rightarrow \nu}} c_\alpha . \end{array} $$

(77)

In the above equation, $R_{b}^{\mu\rightarrow \nu}$ denotes the branch of the tree R _b that is still connected with μ if the directed edge μ→ν is removed; and $R_{j}^{\mu\rightarrow \nu}$ has the same definition, i.e., it is the branch of the tree R _j that contains region μ but not ν. The possibility that a region μ∈B _γ might point to two ore more regions in I _γ does not affect the validity of (77). The reason is simple: if ν ₁ and ν ₂ are two children of μ in I _γ , then ν ₁ and ν ₂ do not share any function node nor any variable node in common.

Based on (77), (76) and (51), we obtain the important expression (52). In that equation, the parent-to-message $m_{\mu\rightarrow \nu}(\underline{x}_{\nu})$ is defined as

$$ m_{\mu\rightarrow \nu}(\underline{x}_\nu) \propto p_{\mu\rightarrow \nu}(\underline{x}_\nu) \prod _{b\in \nu} \bigl[ \psi_b(\underline{x}_{\partial b}) \bigr]^{-\sum_{\alpha\in R_b^{\mu\rightarrow \nu}} c_\alpha} \prod _{j\in \nu} \bigl[ \psi_j(x_j) \bigr]^{-\sum_{\alpha\in R_j^{\mu\rightarrow \nu}} c_\alpha} $$

(78)

up to a normalization constant (to be fixed by $\sum_{\underline{x}_{\nu}} m_{\mu\rightarrow \nu}(\underline{x}_{\nu})=1$).

Using the expression (39) for the probability distribution $p_{\mu\rightarrow \nu}(\underline{x}_{\nu})$, it is easy to show that

(79)

Notice that

(80)

Then we have

(81)

It is easy to check that, for a function node c∈ν∈I _μ ,

$$ \sum _{\alpha \in R_c^{\mu\rightarrow \nu}} c_\alpha - \sum _{\eta \in R_c \cap (I_\mu\backslash I_\nu)} c_\alpha = \sum _{\alpha \in R_c^{\mu\rightarrow \nu} \backslash I_\mu} c_\alpha , $$

(82)

where $R_{c}^{\mu\rightarrow \nu} \backslash I_{\mu}$ denotes the set formed by all the regions in the region subtree $R_{c}^{\mu\rightarrow \nu}$ except those which are also members of the region set I _μ . Similarly, for a variable node k∈ν∈I _μ , we have

$$ \sum _{\alpha \in R_k^{\mu\rightarrow \nu}} c_\alpha - \sum _{\eta \in R_k \cap (I_\mu\backslash I_\nu)} c_\alpha = \sum _{\alpha \in R_k^{\mu\rightarrow \nu} \backslash I_\mu} c_\alpha , $$

(83)

with $R_{k}^{\mu\rightarrow \nu} \backslash I_{\mu}$ being the set formed by all the regions in the region subtree $R_{k}^{\mu\rightarrow \nu}$ except those which are also members of the region set I _μ . With these two equalities, (81) is re-written as

(84)

Combining the above equation with (78) leads to

(85)

Using the properties (77) for the region trees R _b (induced by each function node b∈μ) and R _k (induced by each variable node k∈μ), it is not difficult to verify that the expression in the curly brackets of the above equation is equal to 1. Therefore we arrive at the message-passing equation (53) for $m_{\mu\rightarrow \nu}(\underline{x}_{\nu})$.

Appendix B: Derivation of the Free Energy Expression (56)

We demonstrate that, for a non-redundant region graph R, the region graph free energy F ₀ can be expressed as $F_{0} = \sum_{\alpha \in R} c_{\alpha}\tilde{F}_{\alpha}$, with $\tilde{F}_{\alpha}$ given by (56).

First, we notice that f _(μ,ν) as defined by (42) can be expressed as

$$ f_{(\mu, \nu)} = f_\nu + \frac{1}{\beta} \ln \biggl[ \sum _{\underline{x}_\nu} \varPsi_\nu( \underline{x}_\nu) \prod _{\gamma \in \partial \nu \backslash \mu} p_{\gamma \rightarrow \nu}\bigl(\underline{x}_{\nu\cap \gamma}^{\nu}\bigr) \biggr] = f_\nu - f_{\nu\rightarrow \mu} , $$

(86)

where f _ν→μ is defined through (67). In writing down this equation, we have assumed that μ is a parent of ν. From Eq. (45) we then obtain that

$$ \tilde{F}_\alpha = f_\alpha + \sum _{\{(\mu\rightarrow \nu) | \mu \in I_\alpha\}} f_{\nu\rightarrow \mu} . $$

(87)

On the other hand, based on the definition (42) for f _α we derive that

$$ f_\alpha = -\frac{1}{\beta} \ln \biggl[ \sum _{\underline{x}_\alpha} \prod _{\eta\in I_\alpha} \varPsi_\eta(\underline{x}_\eta) \prod _{\{(\mu\rightarrow \nu) | \mu \in B_\alpha, \nu \in I_\alpha\}} p_{\mu\rightarrow \nu}(\underline{x}_\nu) \biggr] - \sum _{\{(\mu \rightarrow \nu) | \mu \in I_\alpha\}} f_{\nu\rightarrow \mu} . $$

(88)

From the last two expressions we then get the following simple formula for $\tilde{F}_{\alpha}$:

$$ \tilde{F}_\alpha = -\frac{1}{\beta} \ln \biggl[ \sum _{\underline{x}_\alpha} \prod _{\eta\in I_\alpha} \varPsi_\eta(\underline{x}_\eta) \prod _{\{(\mu\rightarrow \nu) | \mu \in B_\alpha, \nu \in I_\alpha\}} p_{\mu\rightarrow \nu}(\underline{x}_\nu) \biggr]. $$

(89)

This formula is very similar to (56), but not yet identical.

Now we replace $p_{\mu\rightarrow \nu}(\underline{x}_{\nu})$ of (89) by $m_{\mu\rightarrow \nu}(\underline{x}_{\nu})$ through the relation (78), and obtain that

(90)

We need to prove that

(91)

To prove this, it is first noticed that the left side of (91) is equivalent to

(92)

For a non-redundant region graph R, the sum in the curly brackets of the above expression is identical to zero, i.e., for each directed edge μ→ν:

$$ \sum _{\{ \alpha | \mu \in B_\alpha, \nu \in I_\alpha\}} c_\alpha = \sum _{\alpha \geq \nu} c_\alpha -\sum _{\eta \geq \mu} c_\eta = 1- 1 = 0 . $$

(93)

Combining Eqs. (44), (89) and (91), we obtain the objective equation:

$$ F_0 = \sum _{\alpha \in R} c_\alpha \biggl\{ -\frac{1}{\beta} \ln \biggl[ \sum _{\underline{x}_\alpha} \prod _{a \in \alpha} \psi_a (\underline{x}_{\partial a}) \prod _{i\in \alpha} \psi_i(x_i) \prod _{\{(\mu\rightarrow \nu) | \mu \in B_\alpha, \nu \in I_\alpha\}} m_{\mu\rightarrow \nu}(\underline{x}_\nu) \biggr] \biggr\} . $$

(94)

Appendix C: Self-consistent Equations at n=2

For the local structure shown in Fig. 6, the square-to-rod messages between the regions α and a are:

(95a)

(95b)

(95c)

where we have introduced several shorthand notations

Similarly, the stripe-to-rod messages between the regions μ and a are:

(96a)

(96b)

(96c)

On the same edges (α,a) and (μ,a), the rod-to-square and rod-to-stripe messages are much simpler and are expressed as

(97a)

(97b)

Appendix D: Stability Analysis of the Paramagnetic Solution at n=2

At the paramagnetic fixed point (72), the effective couplings such as $J_{\alpha\rightarrow a}^{(i j)}$ and $J_{\mu\rightarrow a}^{(i j)}$ are determined self-consistently through (95c) and (96c). Then the rgBP iteration equations for the fields are linearized. The coefficients of the linearized equations are obtained by the following expressions:

(98a)

(98b)

(98c)

(98d)

(98e)

(98f)

(98g)

(98h)

The linearized rgBP equations for the fields are iterated on a given region graph. During each sweep of the iteration, the output field messages of each square region of the region graph are updated once, and the maximum among the absolute values of all the updated fields is recorded. If this maximum decays to zero with the iteration sweeps, the paramagnetic fixed point is then declared as stable. When the paramagnetic solution is unstable, this maximum value will eventually increase with iteration sweeps (after a transient decreasing stage).