scholarly journals Comparison of quantum and classical methods for labels and patterns in Restricted Boltzmann Machines

2021 ◽  
Vol 2122 (1) ◽  
pp. 012007
Author(s):  
Vivek Dixit ◽  
Yaroslav Koshka ◽  
Tamer Aldwairi ◽  
M.A. Novotny
Keyword(s):  
Classification Accuracy ◽  
Quantum Computer ◽  
Quantum Effect ◽  
Low Energy ◽  
Data Reconstruction ◽  
Boltzmann Machine ◽  
Restricted Boltzmann Machines ◽  
Boltzmann Machines ◽  
Comparable Performance ◽  
D Wave

Abstract Classification and data reconstruction using a restricted Boltzmann machine (RBM) is presented. RBM is an energy-based model which assigns low energy values to the configurations of interest. It is a generative model, once trained it can be used to produce samples from the target distribution. The D-Wave 2000Q is a quantum computer which has been used to exploit its quantum effect for machine learning. Bars-and-stripes (BAS) and cybersecurity (ISCX) datasets were used to train RBMs. The weights and biases of trained RBMs were used to map onto the D-Wave. Classification as well as image reconstruction were performed. Classification accuracy of both datasets indicates comparable performance using D-Wave’s adiabatic annealing and classical Gibb’s sampling.

Determination of the Lowest-Energy States for the Model Distribution of Trained Restricted Boltzmann Machines Using a 1000 Qubit D-Wave 2X Quantum Computer

2017 ◽  
Vol 29 (7) ◽  
pp. 1815-1837 ◽  
Author(s):  
Yaroslav Koshka ◽  
Dilina Perera ◽  
Spencer Hall ◽  
M. A. Novotny
Keyword(s):  
Quantum Computer ◽  
Energy State ◽  
Bias Field ◽  
Strength Parameter ◽  
Restricted Boltzmann Machines ◽  
Boltzmann Machines ◽  
Energy States ◽  
Further Development ◽  
D Wave

The possibility of using a quantum computer D-Wave 2X with more than 1000 qubits to determine the global minimum of the energy landscape of trained restricted Boltzmann machines is investigated. In order to overcome the problem of limited interconnectivity in the D-Wave architecture, the proposed RBM embedding combines multiple qubits to represent a particular RBM unit. The results for the lowest-energy (the ground state) and some of the higher-energy states found by the D-Wave 2X were compared with those of the classical simulated annealing (SA) algorithm. In many cases, the D-Wave machine successfully found the same RBM lowest-energy state as that found by SA. In some examples, the D-Wave machine returned a state corresponding to one of the higher-energy local minima found by SA. The inherently nonperfect embedding of the RBM into the Chimera lattice explored in this work (i.e., multiple qubits combined into a single RBM unit were found not to be guaranteed to be all aligned) and the existence of small, persistent biases in the D-Wave hardware may cause a discrepancy between the D-Wave and the SA results. In some of the investigated cases, introduction of a small bias field into the energy function or optimization of the chain-strength parameter in the D-Wave embedding successfully addressed difficulties of the particular RBM embedding. With further development of the D-Wave hardware, the approach will be suitable for much larger numbers of RBM units.

scholarly journals Adaptive hyperparameter updating for training restricted Boltzmann machines on quantum annealers

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Guanglei Xu ◽  
William S. Oates
Keyword(s):  
Neural Network ◽  
Maximum Likelihood ◽  
Image Reconstruction ◽  
Image Recognition ◽  
Shannon Entropy ◽  
Reconstruction Error ◽  
Likelihood Method ◽  
Restricted Boltzmann Machines ◽  
Boltzmann Machines ◽  
D Wave

AbstractRestricted Boltzmann Machines (RBMs) have been proposed for developing neural networks for a variety of unsupervised machine learning applications such as image recognition, drug discovery, and materials design. The Boltzmann probability distribution is used as a model to identify network parameters by optimizing the likelihood of predicting an output given hidden states trained on available data. Training such networks often requires sampling over a large probability space that must be approximated during gradient based optimization. Quantum annealing has been proposed as a means to search this space more efficiently which has been experimentally investigated on D-Wave hardware. D-Wave implementation requires selection of an effective inverse temperature or hyperparameter ($$\beta $$ β ) within the Boltzmann distribution which can strongly influence optimization. Here, we show how this parameter can be estimated as a hyperparameter applied to D-Wave hardware during neural network training by maximizing the likelihood or minimizing the Shannon entropy. We find both methods improve training RBMs based upon D-Wave hardware experimental validation on an image recognition problem. Neural network image reconstruction errors are evaluated using Bayesian uncertainty analysis which illustrate more than an order magnitude lower image reconstruction error using the maximum likelihood over manually optimizing the hyperparameter. The maximum likelihood method is also shown to out-perform minimizing the Shannon entropy for image reconstruction.

scholarly journals Reinforcement learning using quantum Boltzmann machines

2018 ◽  
Vol 18 (1&2) ◽  
pp. 51-74 ◽  
Author(s):  
Daniel Crawford ◽  
Anna Levit ◽  
Navid Ghadermarzy ◽  
Jaspreet S. Oberoi ◽  
Pooya Ronagh
Keyword(s):  
Reinforcement Learning ◽  
Learning Algorithm ◽  
Transverse Field ◽  
Boltzmann Machine ◽  
Quantum Annealing ◽  
Restricted Boltzmann Machines ◽  
Boltzmann Machines ◽  
Ising Spin ◽  
Learning Tasks ◽  
Deep Boltzmann Machine

We investigate whether quantum annealers with select chip layouts can outperform classical computers in reinforcement learning tasks. We associate a transverse field Ising spin Hamiltonian with a layout of qubits similar to that of a deep Boltzmann machine (DBM) and use simulated quantum annealing (SQA) to numerically simulate quantum sampling from this system. We design a reinforcement learning algorithm in which the set of visible nodes representing the states and actions of an optimal policy are the first and last layers of the deep network. In absence of a transverse field, our simulations show that DBMs are trained more effectively than restricted Boltzmann machines (RBM) with the same number of nodes. We then develop a framework for training the network as a quantum Boltzmann machine (QBM) in the presence of a significant transverse field for reinforcement learning. This method also outperforms the reinforcement learning method that uses RBMs.

scholarly journals Analyzing joint brand purchases by conditional restricted Boltzmann machines

2021 ◽  
Author(s):  
Harald Hruschka
Keyword(s):  
Hidden Variables ◽  
Restricted Boltzmann Machine ◽  
Boltzmann Machine ◽  
Restricted Boltzmann Machines ◽  
Market Basket ◽  
Boltzmann Machines ◽  
Independent Variables ◽  
Product Categories ◽  
Pseudo Likelihood ◽  
Marketing Variables

AbstractWe introduce the conditional restricted Boltzmann machine as method to analyze brand-level market basket data of individual households. The conditional restricted Boltzmann machine includes marketing variables and household attributes as independent variables. To our knowledge this is the first study comparing the conditional restricted Boltzmann machine to homogeneous and heterogeneous multivariate logit models for brand-level market basket data across several product categories. We explain how to estimate the conditional restricted Boltzmann machine starting from a restricted Boltzmann machine without independent variables. The conditional restricted Boltzmann machine turns out to excel all the other investigated models in terms of log pseudo-likelihood for holdout data. We interpret the selected conditional restricted Boltzmann machine based on coefficients linking purchases to hidden variables, interdependences between brand pairs as well as own and cross effects of marketing variables. The conditional restricted Boltzmann machine indicates pairwise relationships between brands that are more varied than those of the multivariate logit model are. Based on the pairwise interdependences inferred from the restricted Boltzmann machine we determine the competitive structure of brands by means of cluster analysis. Using counterfactual simulations, we investigate what three different models (independent logit, heterogeneous multivariate logit, conditional restricted Boltzmann machine) imply with respect to the retailer’s revenue if each brand is put on display. Finally, we mention possibilities for further research, such as applying the conditional restricted Boltzmann machine to other areas in marketing or retailing.

scholarly journals Dempster–Shafer Fusion Based on a Deep Boltzmann Machine for Blood Pressure Estimation

2018 ◽  
Vol 9 (1) ◽  
pp. 96 ◽  
Author(s):  
Soojeong Lee ◽  
Joon-Hyuk Chang
Keyword(s):  
Blood Pressure ◽  
Middle Layer ◽  
Upper And Lower Bounds ◽  
Boltzmann Machine ◽  
Restricted Boltzmann Machines ◽  
Boltzmann Machines ◽  
Estimation Uncertainty ◽  
Deep Boltzmann Machine ◽  
Blood Pressure Estimation ◽  
Pressure Estimate

We propose a technique using Dempster–Shafer fusion based on a deep Boltzmann machine to classify and estimate systolic blood pressure and diastolic blood pressure categories using oscillometric blood pressure measurements. The deep Boltzmann machine is a state-of-the-art technology in which multiple restricted Boltzmann machines are accumulated. Unlike deep belief networks, each unit in the middle layer of the deep Boltzmann machine obtain information up and down to prevent uncertainty at the inference step. Dempster–Shafer fusion can be incorporated to enable combined independent estimation of the observations, and a confidence increase for a given deep Boltzmann machine estimate can be clearly observed. Our work provides an accurate blood pressure estimate, a blood pressure category with upper and lower bounds, and a solution that can reduce estimation uncertainty. This study is one of the first to use deep Boltzmann machine-based Dempster–Shafer fusion to classify and estimate blood pressure.

A complete restricted Boltzmann machine on an adiabatic quantum computer

2021 ◽  
pp. 2141003
Author(s):  
Lorenzo Rocutto ◽  
Enrico Prati
Keyword(s):  
Complete Graph ◽  
Quantum Computer ◽  
Restricted Boltzmann Machine ◽  
Computational Time ◽  
Quantum Computers ◽  
Boltzmann Machine ◽  
Computational Power ◽  
Trade Off ◽  
Boltzmann Machines ◽  
The Neural Network

Boltzmann Machines constitute a paramount class of neural networks for unsupervised learning and recommendation systems. Their bipartite version, called Restricted Boltzmann Machine (RBM), is the most developed because of its satisfactory trade-off between computability on classical computers and computational power. Though the diffusion of RBMs is quite limited as their training remains hard. Recently, a renewed interest has emerged as Adiabatic Quantum Computers (AQCs), which suggest a potential increase of the training speed with respect to conventional hardware. Due to the limited number of connections among the qubits forming the graph of existing hardware, associating one qubit per node of the neural network implies an incomplete graph. Thanks to embedding techniques, we developed a complete graph connecting nodes constituted by virtual qubits. The complete graph outperforms previous implementations based on incomplete graphs. Despite the fact that the learning rate per epoch is still slower with respect to a classical machine, the advantage is expected by the increase of number of nodes which impacts on the classical computational time but not on the quantum hardware based computation.

scholarly journals Restricted Boltzmann Machine representation for the groundstate and excited states of Kitaev Honeycomb model

2021 ◽  
Author(s):  
Mohammadreza Noormandipour ◽  
Youran Sun ◽  
Babak Haghighat
Keyword(s):  
Excited States ◽  
Wave Functions ◽  
Boltzmann Machine ◽  
Restricted Boltzmann Machines ◽  
Conformal Blocks ◽  
Boltzmann Machines ◽  
Exact Ground State ◽  
Small Lattice ◽  
Topological Field ◽  
Machine Representation

Abstract In this work, the capability of restricted Boltzmann machines (RBMs) to find solutions for the Kitaev honeycomb model with periodic boundary conditions is investigated. The measured groundstate (GS) energy of the system is compared and, for small lattice sizes (e.g. 3×3 with 18 spinors), shown to agree with the analytically derived value of the energy up to a deviation of 0.09 %. Moreover, the wave-functions we find have 99.89 % overlap with the exact ground state wave-functions. Furthermore, the possibility of realizing anyons in the RBM is discussed and an algorithm is given to build these anyonic excitations and braid them for possible future applications in quantum computation. Using the correspondence between topological field theories in (2+1)d and 2d CFTs, we propose an identification between our RBM states with the Moore-Read state and conformal blocks of the 2 d Ising model.

scholarly journals Inverse problems for structured datasets using parallel TAP equations and restricted Boltzmann machines

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Aurelien Decelle ◽  
Sungmin Hwang ◽  
Jacopo Rocchi ◽  
Daniele Tantari
Keyword(s):  
Inverse Problems ◽  
Efficient Algorithm ◽  
Mean Field ◽  
Boltzmann Machine ◽  
Restricted Boltzmann Machines ◽  
Boltzmann Machines ◽  
Teacher Student ◽  
Pseudo Likelihood ◽  
Field Approaches ◽  
Retrieval Phase

AbstractWe propose an efficient algorithm to solve inverse problems in the presence of binary clustered datasets. We consider the paradigmatic Hopfield model in a teacher student scenario, where this situation is found in the retrieval phase. This problem has been widely analyzed through various methods such as mean-field approaches or the pseudo-likelihood optimization. Our approach is based on the estimation of the posterior using the Thouless–Anderson–Palmer (TAP) equations in a parallel updating scheme. Unlike other methods, it allows to retrieve the original patterns of the teacher dataset and thanks to the parallel update it can be applied to large system sizes. We tackle the same problem using a restricted Boltzmann machine (RBM) and discuss analogies and differences between our algorithm and RBM learning.

Boltzmann machines with clusters of stochastic binary units

2016 ◽  
Vol 07 (02) ◽  
pp. 1650018
Author(s):  
Da Teng ◽  
Zhang Li ◽  
Guanghong Gong ◽  
Liang Han
Keyword(s):  
Gaussian Distribution ◽  
Learning Algorithm ◽  
Hidden Variables ◽  
Recognition Task ◽  
Boltzmann Machine ◽  
Restricted Boltzmann Machines ◽  
Boltzmann Machines ◽  
New Variant ◽  
Deep Boltzmann Machine ◽  
New Learning

The original restricted Boltzmann machines (RBMs) are extended by replacing the binary visible and hidden variables with clusters of binary units, and a new learning algorithm for training deep Boltzmann machine of this new variant is proposed. The sum of binary units of each cluster is approximated by a Gaussian distribution. Experiments demonstrate that the proposed Boltzmann machines can achieve good performance in the MNIST handwritten digital recognition task.

scholarly journals A 64 bit quantum dragon data-set for machine learning

2021 ◽  
Vol 2122 (1) ◽  
pp. 012005
Author(s):  
M.A. Novotný ◽  
Yaroslav Koshka ◽  
G. Inkoonv ◽  
Vivek Dixit
Keyword(s):  
Machine Learning ◽  
Finite Interval ◽  
Tight Binding ◽  
Tight Binding Model ◽  
Restricted Boltzmann Machines ◽  
Binding Model ◽  
Data Set ◽  
Boltzmann Machines ◽  
Electron Transmission ◽  
D Wave

Abstract Design and examples of a sixty-four bit quantum dragon data-set are presented. A quantum dragon is a tight-binding model for a strongly disordered nanodevice, but when connected to appropriate semi-infinite leads has complete electron transmission for a finite interval of energies. The labeled data-set contains records which are quantum dragons, which are not quantum dragons, and which are indeterminate. The quantum dragon data-set is designed to be difficult for trained humans and machines to label a nanodevice with regard to its quantum dragon property. The 64 bit record length allows the data-set to be utilized in restricted Boltzmann machines which fit well onto the D-Wave 2000Q quantum annealer architecture.

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