Accelerate Monte Carlo Simulations with Restricted Boltzmann Machines

Acknowledgments–

L.W. is supported by the Ministry of Science and Technology of China under the Grant No.2016YFA0302400 and the start-up grant of IOP-CAS. L.H. is supported by the Natural Science Foundation of China No.11504340. We acknowledge Jun-Wei Liu, Ye-Hua Liu, Zi-Yang Meng and Yang Qi for useful discussions. We use the Keras library Keras (http://keras.io) is a high level deep learning library based on Theano (http://deeplearning.net/software/theano) and TensorFlow (https://www.tensorflow.org). for training of the neural network and the ALPS library Bauer et al. (2011) for the Monte Carlo data analysis. Our implementation of the restricted Boltzmann machine is based on the scikit-learn library http://scikit-learn.org/stable/modules/generated/sklearn.neural_network.BernoulliRBM.html#.

References

Appendix A Learned weights at T/t=0.13𝑇𝑡0.13T/t=0.13

The learned weights change drastically near the critical temperature. Figure 6 shows the weights learned by the RBM at lower temperature. Compared to Fig. 3 at $T/t=0.15$, there are more hidden neurons controlling extended regions of the visible variables, indicating enlarged correlation length at lower temperature. The checkerboard pattern of the low temperature phase is also more visible.

Appendix B Proof of the detailed balance conditions

We prove the simulation of the RBM shown in Fig. 4 of the main texts satisfies the detailed balance condition.

For the case of block Gibbs sampling shown in Fig. 4(a)

For the case of Gibbs sampler with additional Metropolis steps for the hidden variables shown in Fig. 4(b)

For the second equality we use that the Metropolis update of the hidden variable satisfies $T(\mathbf{h}\rightarrow\mathbf{h}^{\prime})/T(\mathbf{h}^{\prime}\rightarrow\mathbf{h})=p(\mathbf{h}^{\prime})/p(\mathbf{h})$. This proof generalizes to compositions of several of such updates.