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Author: Publisher: ISBN: Category : Languages : en Pages : 27
Book Description
Inference and learning of graphical models are both well-studied problems in statistics and machine learning that have found many applications in science and engineering. However, exact inference is intractable in general graphical models, which suggests the problem of seeking the best approximation to a collection of random variables within some tractable family of graphical models. In this paper, we focus on the class of planar Ising models, for which exact inference is tractable using techniques of statistical physics. Based on these techniques and recent methods for planarity testing and planar embedding, we propose a greedy algorithm for learning the best planar Ising model to approximate an arbitrary collection of binary random variables (possibly from sample data). Given the set of all pairwise correlations among variables, we select a planar graph and optimal planar Ising model defined on this graph to best approximate that set of correlations. Finally, we demonstrate our method in simulations and for two applications: modeling senate voting records and identifying geo-chemical depth trends from Mars rover data.
Author: Publisher: ISBN: Category : Languages : en Pages : 27
Book Description
Inference and learning of graphical models are both well-studied problems in statistics and machine learning that have found many applications in science and engineering. However, exact inference is intractable in general graphical models, which suggests the problem of seeking the best approximation to a collection of random variables within some tractable family of graphical models. In this paper, we focus on the class of planar Ising models, for which exact inference is tractable using techniques of statistical physics. Based on these techniques and recent methods for planarity testing and planar embedding, we propose a greedy algorithm for learning the best planar Ising model to approximate an arbitrary collection of binary random variables (possibly from sample data). Given the set of all pairwise correlations among variables, we select a planar graph and optimal planar Ising model defined on this graph to best approximate that set of correlations. Finally, we demonstrate our method in simulations and for two applications: modeling senate voting records and identifying geo-chemical depth trends from Mars rover data.
Author: Dmitry Kamenetsky Publisher: ISBN: Category : Graphical modeling (Statistics) Languages : en Pages : 242
Book Description
The Ising model is an important model in statistical physics, with over 10,000 papers published on the topic. This model assumes binary variables and only local pairwise interactions between neighbouring nodes. Inference for the general Ising model is NP-hard; this includes tasks such as calculating the partition function, finding a lowest-energy (ground) state and computing marginal probabilities. Past approaches have proceeded by working with classes of tractable Ising models, such as Ising models defined on a planar graph. For such models, the partition function and ground state can be computed exactly in polynomial time by establishing a correspondence with perfect matchings in a related graph. In this thesis we continue this line of research. In particular we simplify previous inference algorithms for the planar Ising model. The key to our construction is the complementary correspondence between graph cuts of the model graph and perfect matchings of its expanded dual. We show that our exact algorithms are effective and efficient on a number of real-world machine learning problems. We also investigate heuristic methods for approximating ground states of non-planar Ising models. We show that in this setting our approximative algorithms are superior than current state-of-the-art methods.
Author: Mikhail Dudalev Publisher: ISBN: Category : Languages : en Pages : 246
Book Description
In this thesis, we study the critical behaviour of the two-dimensional Ising model on the regular lattices. Using numerical solution of the model on the square, triangular and honeycomb lattices we compute the universal scaling function, which turns out to be identical on each of the lattices, in addition to being identical to the scaling function of the Ising Field Theory, computed previously by Fonseca and Zamolodchikov. To cope with the lattice contributions we carefully examined series expansions of the lattice free energy derivatives. We included the non-scaling regular part of the free energy as well as non-linear Aharony-Fisher scaling fields, which all have non-universal expansions. Using as many of the previously known exacts results as possible, we were able to fit the unknown coefficients of the scaling function expansion and obtain some non-universal coefficients. In contrast to the IFT approach of Fonseca and Zamolodchikov, all coefficients were obtained independently from the separate datasets, without using dispersion relations. These results show that the Scaling and Universality hypotheses, with the help of the Aharony-Fisher corrections, hold on the lattice to very high precision and so there should be no doubt of their validity. For all numerical computations we used the Corner Transfer Matrix Renormalisation Group (CTMRG) algorithm, introduced by Nishino and Okunishi. The algorithm combines Baxter's variational approach (which gives Corner Transfer Matrix (CTM) equations), and White's Density Matrix Renormalisation Group (DMRG) method to solve the CTM equations efficiently. It was shown that given sufficient distance from the critical point, the algorithmic precision is exceptionally good and can unlikely be overcome with any other general algorithm using the same amount of numerical computations. While performing tests we also confirmed several critical parameters of the three-state Ising and Blume-Capel models, although no extra precision was gained, compared to previous results from other methods. In addition to the results presented here, we produced an efficient and reusable implementation of the CTMRG algorithm, which after minor modifications could be used for a variety of lattice models, such as the Kashiwara-Miwa and the chiral Potts models. -- provided by Candidate.
Author: Boris Kryzhanovsky Publisher: Springer ISBN: 3030013286 Category : Technology & Engineering Languages : en Pages : 353
Book Description
This book describes new theories and applications of artificial neural networks, with a special focus on addressing problems in neuroscience, biology and biophysics and cognitive research. It covers a wide range of methods and technologies, including deep neural networks, large-scale neural models, brain–computer interface, signal processing methods, as well as models of perception, studies on emotion recognition, self-organization and many more. The book includes both selected and invited papers presented at the XX International Conference on Neuroinformatics, held in Moscow, Russia on October 8–12, 2018.
Author: Publisher: Elsevier ISBN: 0444641416 Category : Mathematics Languages : en Pages : 706
Book Description
Processing, Analyzing and Learning of Images, Shapes, and Forms: Part 2, Volume 20, surveys the contemporary developments relating to the analysis and learning of images, shapes and forms, covering mathematical models and quick computational techniques. Chapter cover Alternating Diffusion: A Geometric Approach for Sensor Fusion, Generating Structured TV-based Priors and Associated Primal-dual Methods, Graph-based Optimization Approaches for Machine Learning, Uncertainty Quantification and Networks, Extrinsic Shape Analysis from Boundary Representations, Efficient Numerical Methods for Gradient Flows and Phase-field Models, Recent Advances in Denoising of Manifold-Valued Images, Optimal Registration of Images, Surfaces and Shapes, and much more. Covers contemporary developments relating to the analysis and learning of images, shapes and forms Presents mathematical models and quick computational techniques relating to the topic Provides broad coverage, with sample chapters presenting content on Alternating Diffusion and Generating Structured TV-based Priors and Associated Primal-dual Methods
Author: David Barber Publisher: Cambridge University Press ISBN: 0521518148 Category : Computers Languages : en Pages : 739
Book Description
A practical introduction perfect for final-year undergraduate and graduate students without a solid background in linear algebra and calculus.
Author: Xue-Cheng Tai Publisher: North Holland ISBN: 0444641408 Category : Languages : en Pages : 704
Book Description
Processing, Analyzing and Learning of Images, Shapes, and Forms: Part 2, Volume 20, surveys the contemporary developments relating to the analysis and learning of images, shapes and forms, covering mathematical models and quick computational techniques. Chapter cover Alternating Diffusion: A Geometric Approach for Sensor Fusion, Generating Structured TV-based Priors and Associated Primal-dual Methods, Graph-based Optimization Approaches for Machine Learning, Uncertainty Quantification and Networks, Extrinsic Shape Analysis from Boundary Representations, Efficient Numerical Methods for Gradient Flows and Phase-field Models, Recent Advances in Denoising of Manifold-Valued Images, Optimal Registration of Images, Surfaces and Shapes, and much more. Covers contemporary developments relating to the analysis and learning of images, shapes and forms Presents mathematical models and quick computational techniques relating to the topic Provides broad coverage, with sample chapters presenting content on Alternating Diffusion and Generating Structured TV-based Priors and Associated Primal-dual Methods
Author: Trevor Hastie Publisher: CRC Press ISBN: 1498712177 Category : Business & Economics Languages : en Pages : 354
Book Description
Discover New Methods for Dealing with High-Dimensional DataA sparse statistical model has only a small number of nonzero parameters or weights; therefore, it is much easier to estimate and interpret than a dense model. Statistical Learning with Sparsity: The Lasso and Generalizations presents methods that exploit sparsity to help recover the underl
Author: Iwao Teraoka Publisher: John Wiley & Sons ISBN: 1119375282 Category : Science Languages : en Pages : 382
Book Description
This textbook introduces chemistry and chemical engineering students to molecular descriptions of thermodynamics, chemical systems, and biomolecules. Equips students with the ability to apply the method to their own systems, as today's research is microscopic and molecular and articles are written in that language Provides ample illustrations and tables to describe rather difficult concepts Makes use of plots (charts) to help students understand the mathematics necessary for the contents Includes practice problems and answers
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Author: Dmitry Kamenetsky
Author: Mikhail Dudalev
Author: Boris Kryzhanovsky
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Author: David Barber
Author: Xue-Cheng Tai
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Author: Trevor Hastie
Author: Iwao Teraoka