where Z=∑{v,h}e−E(v,h) is the canonical partition function. S. Ioffe and C. Szegedy, arXiv Our work complements earlier studies that utilized one RBM per temperature to learn the distribution of Ising configurations with a few dozen spins Torlai and Melko (2016); Morningstar and Melko (2017); Iso et al. (2009). IV.1 and in Sec. the 25th international conference on Machine learning, Proceedings of (2019). on Information Theory. the thirteenth international conference on artificial intelligence and In Secs. representation learning. The sum ∑{v,h} is taken over all possible configurations {v,h}. Assoc. B.1 shows the distribution of E(T) and M(T) in the original data. The left panel of Fig. ∙ For further details on the RBM training process, see Sec. Join one of the world's largest A.I. Physical features in non-convolutional VAE Ising samples. The ELBO is the loss function of a VAE. share, Generative models of 3D human motion are often restricted to a small num... B.1. IV, we apply RBMs and VAEs to learn the distribution of Ising spin configurations for different temperatures. Since the true posterior p(z|x) is unknown, we cannot solve the optimization problem of Eq. dependence of magnetization, energy, and spin-spin correlations. To generate samples from an RBM, we start 200 Markov chains from random initial configurations whose mean corresponds to the mean of spins in the data at the desired temperature. In Fig. where ~x corresponds to the vector x but with a flipped i-th component. ∙ ETH Zurich ∙ 0 ∙ share This week in AI Get the week's most popular data science and artificial intelligence research ∙ To better visualize the differences between original data and model-generated Ising samples, we show two-dimensional scatter plots of energy and magnetization in App. %PDF-1.5 P. Smolensky, Parallel Distributed Processing: Explorations in the Microstructure of ∙ share, Here we introduce a new model of natural textures based on the feature s... In addition to using one RBM per temperature, we also train one RBM for all temperatures. We may express the log-likelihood function in terms of the free energy, To train an RBM, we have to find a set of parameters θ that minimizes the difference between the distribution pθ of the machine and the distribution ^p of the data. RBMs are a particular type of stochastic neural networks and were first introduced by Smolensky in 1986 under the name Harmonium Smolensky . and S. Verdú, IEEE Transactions Beach, and R. G. Melko, Phys. Res. A.1, we show the resulting temperature dependence of magnetization, energy, and spin correlations. In particular, restricted Boltzmann machines (RBMs) and where xi∈{0,1} and pi(x) is the reconstruction probability of bit i. This specific initialization together with the encoding of corresponding temperature labels helps sampling from single RBMs that were trained for all temperatures. IV.2, we also considered their non-convolutational counterparts. 01/15/2020 ∙ by Francesco D'Angelo, et al. Our results suggest that ∙ These modified distributions are used in the ELBO of a cVAE. This optimization problem can be solved using a stochastic gradient descent algorithm, denotes the loss function (i.e., negative log-likelihood or KL divergence) and. We denote the parameter sets of the encoder and decoder by ϕ and θ, . %� share. Our paper proceeds as follows. Due to the absence of intra-layer connections in RBMs, hidden (visible) variables are mutually independent given the visible (hidden) ones. Phys. where ϵ∼N(0,I) and ⊙ is the element-wise product. During training, we keep track of the ELBO (see Eq. P. Mehta, M. Bukov, (8) and (16)). Group Flow, Texture Synthesis Using Convolutional Neural Networks, Restricted Boltzmann Machine Flows and The Critical Temperature of Ising statistics. Despite these successes, however, there is only For the cVAE, we sample from the decoder using 20×103 random normal 200-dimensional vectors for each temperature. 09/03/2020 ∙ by Sascha Diefenbacher, et al. An alternative to CD-k methods is persistent contrastive divergence (PCD) Tieleman (2008). (15) directly and instead have to maximize the evidence lower bound (ELBO) Kingma and Welling (2013): where p(z)=N(0,I) is the latent variable prior. 0 In previous studies, RBMs were applied to smaller systems with only about 100 spins to learn the distribution of spin configurations with one RBM per temperature (see Refs. However, samples generated by the employed VAE are less evenly distributed across temperatures than is the case for samples that we generated with an RBM. Significant advances in deep learning have led to more widely used and The first term in Eq. 0 For the decoder, we use an input layer that consists of 200 units to represent the latent variable z and concatenate it with the additional temperature labels. In addition to the convolutional architecture of cVAEs that we described in Sec. Each convolutional layer is followed by a. We also consider non-convolutional VAE architectures in App. 03/20/2018 ∙ by Alexander Sagel, et al. We first focus on the training of RBMs in Sec. S. Efthymiou, M. J. << /Filter /FlateDecode /Length 3532 >> Furthermore, we needed to use a convolutational architecure for VAEs to capture spin correlations and energy. realizations {x1,…,xN}. Physical features in RBM and convolutional VAE Ising samples. The antiferromagnetic Ising model has a negative interaction parameter, whereby neighbor- ing nodes prefer to be in opposite states. 7, we show some snapshots of Ising configurations in the original data and compare them to spin configurations, which we generated with a single RBM and convolutional cVAE. In the case of single RBMs and cVAEs that we train for all temperatures, we have to use a classifier to determine the temperature of the generated samples. preprint arXiv:1502.03167 (2015). In contrast to Boltzmann machines Ackley et al. The output layer is linear and consists of 1024 units. II to learn the distribution of Ising configurations at different temperatures. The idea behind PCD is that the model distribution only changes slightly after updating the model parameters θ according to Eq. We conclude our study and discuss our results in Sec. We find that the RBMs we consider have problems to reproduce the behavior at high temperatures (e.g., for T=4, see middle panel of Fig. p... IV.1. 13, 12 and 11. Rep. (2019). share, Spin-glass systems are universal models for representing many-body pheno... In the following subsections, we describe additional methods that allow us to monitor approximations of log-likelihood (see Eq. and J. Nocedal, SIAM Rev. j��ǎ��z�s����q�������cUL� ��#���Ы�������F���Wxn8�)�}�r>Y=U�x�i�G��/�U-�ҪX�_-7�T3��5^L!�(�Go�OSU׆;�T��� �����?���%�������\j���X��p�8���o�E�0�R?O�[�)�r��Z��{mF~��Ӫmg5���:���*����'I?�4����(? Both models have in common that they create a low-dimensional representation of a given dataset and restore them minimizing the error in the reconstruction (see Eqs. The second neural network in a VAE is a decoder that uses samples of the Gaussian qϕ(z|x) as input to generate new samples according to a distribution pθ(x|z) (see Fig. After determining the temperatures of the generated samples (see Sec. (2007). Furthermore, we determine the spin-spin correlations between two spins σi and σj at positions ri and rj: We use translational-invariance and rewrite Eq. We observe that energy and correlations are not captured as well as for convolutational architectures and RBMs (see Fig. The bipartite network structure of RBMs allows to update units within one layer in parallel (see Eqs. Mach. Am. To better understand the representational characteristics of RBMs and VAEs, we studied the ability of these models to learn the distribution of physical features of Ising configurations at different temperatures. We repeated this procedure 20 times for each temperature and averaged over minibatches of 256 samples to have better estimators. D. P. Kingma and M. Welling, arXiv One possibility to monitor the learning progress of an RBM is the so-called pseudo log-likelihood Besag (1975). This approach can only provide limited insights into the representational properties of neural network models. 06/17/2020 ∙ by Rodrigo Veiga, et al. L. Bottou, F. E. Curtis, The Ising model is a renowned model in statistical physics that was originally introduced to study the phase transition phenomenon in ferromagnetic materials (1). All codes and further detailed descriptions are available on GitHub git . We use the same samples and training parameters as before. ∙ where r=|ri−rj| is the distance between spins σi and σj. Next, we determine magnetization, energy, and correlation functions of the generated samples at different temperatures (see Fig. We also consider the training of one RBM per temperature and compare the results to the samples generated with an RBM and a cVAE that were trained for all temperatures. Instead we use supervised learning to train a simple Neural Network to automagically learn the transition temperature. We estimate all parameters by maximizing the ELBO using backpropagation, . In comparison with other generative neural networks, the training of RBMs is challenging and computationally expensive Schulz et al.
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