Transforming generalized Ising models into Boltzmann machines

We find an exact mapping from the generalized Ising models with many-spin interactions to equivalent Boltzmann machines, i.e., the models with only two-spin interactions between physical and auxiliary binary variables accompanied by local external fields. More precisely, the appropriate combination of the algebraic transformations, namely the star-triangle and decoration-iteration transformations, allows one to express the model in terms of fewer-spin interactions at the expense of the degrees of freedom. Furthermore, the benefit of the mapping in Monte Carlo simulations is discussed. In particular, we demonstrate that the application of the method in conjunction with the Swendsen-Wang algorithm drastically reduces the critical slowing down in a model with two- and three-spin interactions on the Kagomé lattice.

Figure 1

Schematic picture of (a) restricted Boltzmann machine (RBM) that is equivalent to a model with three-spin Ising-type interaction, (b) deep Boltzmann machine (DBM) that is equivalent to a model with four-spin Ising-type interaction. The white and black objects denote visible and deep spins, respectively. Also the blue and red objects are the hidden spins introduced by the decoration-iteration transformation (DIT) and star-triangle transformation (STT), which are given by Eqs. (5) and (7). The presence of the layers is denoted by the gray planes. Note that the mapping to the RBM or DBM is applicable irrespective of the spatial dimension.

Figure 2

Schematic description of the mapping techniques: (a) Decoration-iteration transformation and (b) star-triangle transformation. The white and black circles correspond the visible and hidden spins, respectively, and the numbers denote the labels of the visible spins. The solid lines and the filled region denote the two- and three-spin interactions, respectively.

Figure 3

(a) Transforming a pure three-spin interaction into an RBM. The red filled circles in the right-hand side are generated by the STT and the blue ones by the DIT. The black solid and dotted lines denote the positive and negative two-spin interactions, respectively. The amplitudes of the virtual two-spin interactions are taken as $J_i>0$ in the figure. (b) Transforming a couple of three-spin interactions. The signs of the virtual two-spin interactions are modified so that the number of the hidden spins is reduced. (c) Transforming the Baxter-Wu model into an RBM.

Figure 4

(a) Transforming four-spin interaction into a DBM. The black filled circle is the spin in the deep layer and the other notation follows that of Fig. 3. (b) Transforming $p$-spin interaction into a DBM. The architecture of the BM is determined from the number of the spins involved in the interaction, but is irrelevant to the actual spatial dimension.

Figure 5

Graphical understanding of the transformation for the spin model defined by Eq. (26) into RBM and hidden-spin-only model. The original space is defined on the Kagomé lattice, in which the triangles are colored to denote the signs of three-spin interactions. The white open and red filled circles correspond to the visible and hidden spins, respectively. Also the signs of the two-spin interactions are represented as solid and dotted lines for positive and negative, respectively.

Figure 6

Data collapse of the Binder ratio for the Ising model with two-spin and alternating-sign three-spin interaction on the Kagomé lattice. The critical temperature and the critical exponent for the correlation length are obtained as $T_{\mathrm{c}}\sim 2.141$ and $\nu \sim 0.99$, respectively. The magnitude of two-spin interactions is set to unity while that of three-spin interactions is $M/{\beta }_{\mathrm{c}}$ = 0.1. The blue circles, green upward triangles, red squares, and purple downward triangles denote the data for the linear system sizes $L=12,18,24,$ and 36, respectively.

Figure 7

(a) The autocorrelation time of the magnetization measured in units of Monte Carlo step. The black circle, purple upward triangle, red rectangle, green downward triangle, and blue diamond markers denote the magnitude of three-spin interactions to be $0,0.05,0.1,0.2,$ and 0.5, respectively. The filled (unfilled) markers represent the results by the single-spin flip in the original space (cluster update in the extended space). (b) The phase diagram of the model defined in Eq. (26). The boundary between the ferromagnetic (FM) and the paramagnetic (PM) phases are given here. The blue dashed line is calculated from the approximate solution in Ref. [63], and the black dots are given by the finite-size scaling of the Binder ratio. The numerically estimated inverse critical temperature at $M=0$ approaches ${\beta }_{\mathrm{c}}=ln\left(3+2\sqrt{3}\right)/4\sim 0.4666$, which can be obtained from the exact solution [40].

Figure 8

The transformation of (a) five-spin interaction and (b) $p$-spin interaction. First mapped into a system with only three-spin interactions by applying DIT, the transformation technique for the three-spin interaction is used to break up into two-spin interactions with local external fields.