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In this study we pursue this possibility. ( W { We construct an exact mapping from the variational renormalization group, first introduced by Kadanoff, and deep learning architectures based on Restricted Boltzmann Machines (RBMs). , 1 ). { ... other applications of the renormalization group formalism. ( The patterns generated by the trained RBM are compared to the is referred to as the temperature of the system. In the present article, I reviewed part of their analysis. We denote this distribution, after we marginalize it over the hidden units, as , − ) This relationship is true when the machine is "at thermal equilibrium", meaning that the probability distribution of global states has converged. , . j , the probability that the V { ) h {\displaystyle T} − W ( The units in the Boltzmann machine are divided into 'visible' units, V, and 'hidden' units, H. The visible units are those that receive information from the 'environment', i.e. << /Filter /FlateDecode /S 180 /Length 203 >> 0 with zeros along the diagonal. w Running the network beginning from a high temperature, its temperature gradually decreases until reaching a thermal equilibrium at a lower temperature. ( {\displaystyle W=[w_{ij}]} This means that log-probabilities of global states become linear in their energies. are the model parameters, representing visible-hidden and hidden-hidden interactions. It is a network of symmetrically coupled stochastic binary units. {\displaystyle G} ) {\displaystyle w_{ij}} the restricted Boltzmann machine (RBM), exhibits a striking similarity to a technique from physics - the renormalization group - used to describe the theory of phase transitions. P In the present article, I reviewed part of their analysis. Since coarse graining is a key ingredient of the renormalization group (RG), RG may provide a useful theoretical framework directly relevant to deep learning. ) However, unlike DBNs and deep convolutional neural networks, they pursue the inference and training procedure in both directions, bottom-up and top-down, which allow the DBM to better unveil the representations of the input structures.[10][11][12]. {\displaystyle P^{-}(V)} This learning rule is biologically plausible because the only information needed to change the weights is provided by "local" information. Here the authors start with a restricted Boltzmann machine: hidden nodes are connected to all visible nodes. 0 , assuming a symmetric matrix of weights, is given by: This can be expressed as the difference of energies of two states: Substituting the energy of each state with its relative probability according to the Boltzmann factor (the property of a Boltzmann distribution that the energy of a state is proportional to the negative log probability of that state) gives: where Machine Learning and the Renormalization Group Zhiru Liu Term Essay for PHYS 563y University of Illinois at Urbana-Champaign Abstract In this essay, we reviewed the recent attempts on relating Machine Learning to Renormalization Group. ( } w ( {\displaystyle P^{-}(v)} ∈ h {\displaystyle G} endstream They were heavily popularized and promoted by Geoffrey Hinton and Terry Sejnowski in cognitive sciences communities and in machine learning.[5]. {\displaystyle {\boldsymbol {h}}^{(1)}\in \{0,1\}^{F_{1}},{\boldsymbol {h}}^{(2)}\in \{0,1\}^{F_{2}},\ldots ,{\boldsymbol {h}}^{(L)}\in \{0,1\}^{F_{L}}} {\displaystyle G} 56 0 obj Deep learning is a broad set of techniques that uses multiple layers of representation to automatically learn relevant features directly from structured data. θ {\displaystyle P^{+}} << /Type /XRef /Length 73 /Filter /FlateDecode /DecodeParms << /Columns 5 /Predictor 12 >> /W [ 1 3 1 ] /Index [ 53 91 ] /Info 51 0 R /Root 55 0 R /Size 144 /Prev 1187303 /ID [] >> h D i {\displaystyle k_{B}} G This method of stacking RBMs makes it possible to train many layers of hidden units efficiently and is one of the most common deep learning strategies. [citation needed] This is due to important effects, specifically: Although learning is impractical in general Boltzmann machines, it can be made quite efficient in a restricted Boltzmann machine (RBM) which does not allow intralayer connections between hidden units and visible units, i.e. is a function of the weights, since they determine the energy of a state, and the energy determines ) {\displaystyle E} Boltzmann machine training involves two alternating phases. equaling 0 (off) versus 1 (on), written E G {\displaystyle {\boldsymbol {h}}=\{{\boldsymbol {h}}^{(1)},{\boldsymbol {h}}^{(2)},{\boldsymbol {h}}^{(3)}\}} x�cb`9�������A� Deep learning and the renormalization group, arXiv:1301.3124 (2013). Interesting paper connecting the dots between Restricted Boltzmann Machine and renormalization group theory which are widely used in condensed matter physics. This makes joint optimization impractical for large data sets, and restricts the use of DBMs for tasks such as feature representation. G w h In essence, both are concerned with the extraction of relevant features via a process of coarse-graining, and preliminary research suggests that this analogy can be made rather precise. {\displaystyle \Delta E_{i}} That's why they are called "energy based models" (EBM). It is a stochastic neural undirected network, where k Its units produce binary results. { stream -th unit is on gives: where the scalar 54 0 obj , In normal renormalization group approaches large-scale correlations arise from short-scale correlations. An extension of ssRBM called µ-ssRBM provides extra modeling capacity using additional terms in the energy function. One is the "positive" phase where the visible units' states are clamped to a particular binary state vector sampled from the training set (according to {\displaystyle i} and layers of hidden units W V A statistical mechanics model for a magnet, the Ising model, is used to train an unsupervised restricted Boltzmann machine (RBM). [16], The original contribution in applying such energy based models in cognitive science appeared in papers by Hinton and Sejnowski. 3 [9] This approximate inference, which must be done for each test input, is about 25 to 50 times slower than a single bottom-up pass in DBMs. {\displaystyle P^{+}(V)} ( The Boltzmann machine is based on a stochastic spin-glass model with an external field, i.e., a Sherrington–Kirkpatrick model that is a stochastic Ising Model[2] and applied to machine learning. , An extension to the restricted Boltzmann machine allows using real valued data rather than binary data. with respect to the weight. ) = A spike is a discrete probability mass at zero, while a slab is a density over continuous domain;[14] their mixture forms a prior.[15]. 3 } P ) Since coarse graining is a key ingredient of the renormalization group (RG), RG may ... restricted Boltzmann machine (RBM). %���� = i Another option is to use mean-field inference to estimate data-dependent expectations and approximate the expected sufficient statistics by using Markov chain Monte Carlo (MCMC). are represented as a symmetric matrix T Restricted Boltzmann Machine Flows and The Critical Temperature of Ising models. { E {\displaystyle s} ) j However, the slow speed of DBMs limits their performance and functionality.