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Feature extraction of two dimensional Ising model by unsupervised neural networks

純斗 横尾, スミト ヨコオ, Sumito Yokoo

Published Mar 24, 2020• 0 citations
OpenAlex

Abstract

Deep Neural Network (DNN) has been very successful.Especially, unsupervised DNN can extract the features of complicated dataset without any prior knowledge.However, theoretical understanding of how it extracts the features and what it extracts as the features is not clear.We investigate these questions from the theoretical physics point of view by using the Restricted Boltzmann Machine (RBM) and the Autoencoder (AE) trained by spin configurations of the Ising model.In order to explore the features RBM and AE extract, we train them by spin configurations of the 2-dimensional Ising model and construct a flow of its parameters (in particular temperature) generated by the trained RBM and AE.This flow is motivated by the renormalization group flow of statistical model.It is thought that this flow emphasizes the "relevant" features the unsupervised networks learn, and eliminate "irrelevant" information from the dataset.We show that the RBM trained by spin configurations at various temperatures generates a flow on which the temperature approaches the critical temperature T c ≃ 2.27 if the number of visible units is less than hidden units.Unlike the RBM case, the temperature of the fixed points of AE trained sufficiently is lower than the critical temperature if AE is sufficiently trained (5000 epochs).On the other hand, if the learning epochs are 1000, the fixed point temperature monotonically increase with the number of hidden units.We also analyze the weight matrices by singular value decomposition in order to read out the information RBM and AE learn.4 Autoencoder 4.1 Setup . . . . . .

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Deep Neural Network (DNN) has been very successful.Especially, unsupervised DNN can extract the features of complicated dataset without any prior knowledge.However, theoretical understanding of how it extracts the features and what it extracts as the features is not clear.We investigate these questions from the theoretical physics point of view by using the Restricted Boltzmann Machine (RBM) and the Autoencoder (AE) trained by spin configurations of the Ising model.In order to explore the features RBM and AE extract, we train them by spin configurations of the 2-dimensional Ising model and construct a flow of its parameters (in particular temperature) generated by the trained RBM and AE.This flow is motivated by the renormalization group flow of statistical model.It is thought that this flow emphasizes the "relevant" features the unsupervised networks learn, and eliminate "irrelevant" information from the dataset.We show that the RBM trained by spin configurations at various temperatures generates a flow on which the temperature approaches the critical temperature T c ≃ 2.27 if the number of visible units is less than hidden units.Unlike the RBM case, the temperature of the fixed points of AE trained sufficiently is lower than the critical temperature if AE is sufficiently trained (5000 epochs).On the other hand, if the learning epochs are 1000, the fixed point temperature monotonically increase with the number of hidden units.We also analyze the weight matrices by singular value decomposition in order to read out the information RBM and AE learn.4 Autoencoder 4.1 Setup . . . . . .
Theoretical understanding of how a deep neural network (DNN) extracts features from input images is still unclear, but it is widely believed that the extraction is performed hierarchically through a process of coarse graining. It reminds us of the basic renormalization group (RG) concept in statistical physics. In order to explore possible relations between DNN and RG, we use the restricted Boltzmann machine (RBM) applied to an Ising model and construct a flow of model parameters (in particular, temperature) generated by the RBM. We show that the unsupervised RBM trained by spin configurations at various temperatures from T=0 to T=6 generates a flow along which the temperature approaches the critical value T_{c}=2.27. This behavior is the opposite of the typical RG flow of the Ising model. By analyzing various properties of the weight matrices of the trained RBM, we discuss why it flows towards T_{c} and how the RBM learns to extract features of spin configurations.
We study the features extracted by the Restricted Boltzmann Machine (RBM) when it is trained with spin configurations of Ising model at various temperatures. Using the trained RBM, we obtain the flow of iterative reconstructions (RBM flow) of the spin configurations and find that in some cases the flow approaches the phase transition point T=TcT=T_cT=Tc​ in Ising model. Since the extracted features are emphasized in the reconstructed configurations, the configurations at such a fixed point should describe nothing but the extracted features. Then we investigate the dependence of the fixed point on various parameters and conjecture the condition where the fixed point of the RBM flow is at the phase transition point. We also provide supporting evidence for the conjecture by analyzing the weight matrix of the trained RBM.
Machine learning methods are powerful in distinguishing different phases of matter in an automated way and provide a new perspective on the study of physical phenomena.We train a restricted Boltzmann machine (RBM) on data constructed with spin configurations sampled from the Ising Hamiltonian at different values of temperature and external magnetic field using Monte Carlo methods.From the trained machine we obtain the flow of iterative reconstruction of spin state configurations to faithfully reproduce the observables of the physical system.We find that the flow of the trained RBM approaches the spin configurations of the maximal possible specific heat which resemble the near-criticality region of the Ising model.In the special case of the vanishing magnetic field the trained RBM converges to the critical point of the renormalization group (RG) flow of the lattice model.Our results suggest an explanation of how the machine identifies the physical phase transitions, by recognizing certain properties of the configuration like the maximization of the specific heat, instead of associating directly the recognition procedure with the RG flow and its fixed points.Then from the reconstructed data we deduce the critical exponent associated with the magnetization to find satisfactory agreement with the actual physical value.We assume no prior knowledge about the criticality of the system and its Hamiltonian.
We explore alternative experimental setups for the iterative sampling (flow) from Restricted Boltzmann Machines (RBM) mapped on the temperature space of square lattice Ising models by a neural network thermometer. This framework has been introduced to explore connections between RBM-based deep neural networks and the Renormalization Group (RG). It has been found that, under certain conditions, the flow of an RBM trained with Ising spin configurations approaches in the temperature space a value around the critical one: $ k_B T_c / J \approx 2.269$. In this paper we consider datasets with no information about model topology to argue that a neural network thermometer is not an accurate way to detect whether the RBM has learned scale invariance or not.
The restricted Boltzmann machine (RBM) is a two-layer energy-based model that uses its hidden-visible connections to learn the underlying distribution of visible units, whose interactions are often complicated by high-order correlations. Previous studies on the Ising model of small system sizes have shown that RBMs are able to accurately learn the Boltzmann distribution and reconstruct thermal quantities at temperatures away from the critical point Tc. How the RBM encodes the Boltzmann distribution and captures the phase transition are, however, not well explained. In this work, we perform RBM learning of the 2d and 3d Ising model and carefully examine how the RBM extracts useful probabilistic and physical information from Ising configurations. We find several indicators derived from the weight matrix that could characterize the Ising phase transition. We verify that the hidden encoding of a visible state tends to have an equal number of positive and negative units, whose sequence is randomly assigned during training and can be inferred by analyzing the weight matrix. We also explore the physical meaning of the visible energy and loss function (pseudo-likelihood) of the RBM and show that they could be harnessed to predict the critical point or estimate physical quantities such as entropy.
The renormalization group (RG) is an essential technique in statistical physics and quantum field theory, which considers scale-invariant properties of physical theories and how these theories' parameters change with scaling. Deep learning is a powerful computational technique that uses multi-layered neural networks to solve a myriad of complicated problems. Previous research suggests the possibility that unsupervised deep learning may be a form of RG flow, by being a layer-by-layer coarse graining of the original data. We examined this connection on a more rigorous basis for the simple example of Kadanoff block renormalization of the 2D nearest-neighbor Ising model, with our deep learning accomplished via Restricted Boltzmann Machines (RBMs). We developed extensive renormalization techniques for the 1D and 2D Ising model to provide a baseline for comparison. For the 1D Ising model, we successfully used Adam optimization on a correlation length loss function to learn the group flow, yielding results consistent with the analytical model for infinite N. For the 2D Ising model, we successfully generated Ising model samples using the Wolff algorithm, and performed the group flow using a quasi-deterministic method, validating these results by calculating the critical exponent ν. We then examined RBM learning of the Ising model layer by layer, finding a blocking structure in the learning that is qualitatively similar to RG. Lastly, we directly compared the weights of each layer from the learning to Ising spin renormalization, but found quantitative inconsistencies for the simple case of nearest-neighbor Ising models.