Monte Carlo methods:
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Springer
[2020]
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| Schlagworte: | |
| Online-Zugang: | Inhaltstext http://www.springer.com/ Inhaltsverzeichnis |
| Beschreibung: | xvi, 422 Seiten Ilustrationen; Diagramme 23.5 cm x 15.5 cm |
| ISBN: | 9789811329708 9811329702 |
Internformat
MARC
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Datensatz im Suchindex
| _version_ | 1804181193744187392 |
|---|---|
| adam_text | Contents 1 Introduction to Monte Carlo Methods ............................................................. 1.1 Introduction.................................................................................................. 1.2 Motivation and Objectives........................................................................... 1.3 Tasks in Monte Carlo Computing................................................................ 1.3.1 Task 1: Sampling and Simulation............................................... 1.3.2 Task 2: Estimating Quantities by Monte Carlo Simulation...... 1.3.3 Task 3: Optimization and Bayesian Inference........................... 1.3.4 Task 4: Learning and Model Estimation..................................... 1.3.5 Task 5: Visualizing the Landscape ............................................ References............................................................................................................... 1 1 2 3 4 7 9 11 12 16 2 Sequential MonteCarlo ........................................................................................ 2.1 Introduction.................................................................................................. 2.2 Sampling a 1-Dimensional Density........................................................... 2.3 Importance Sampling and Weighted Samples........................................... 2.4 Sequential Importance Sampling (SIS)..................................................... 2.4.1 Application: The Number of Self-Avoiding Walks..................... 2.4.2 Application: Particle Filtering for Tracking
Objects in a Video....................................................................................... 2.4.3 Summary of the SMC Framework.............................................. 2.5 Application: Ray Tracing by SMC............................................................. 2.5.1 Example: Glossy Highlights........................................................ 2.6 Preserving Sample Diversity in Importance Sampling............................. 2.6.1 Parzen Window Discussion.......................................................... 2.7 Monte Carlo Tree Search............................................................................ 2.7.1 Pure Monte Carlo Tree Search.................................................... 2.7.2 AlphaGo........................................................................................ 2.8 Exercises...................................................................................................... References............................................................................................................... 19 19 19 20 24 25 28 31 33 34 35 39 41 42 44 46 48 vii
viii Contents 3 Markov Chain Monte Carlo: The Basics.......................................................... 3.1 Introduction.................................................................................................. 3.2 Markov Chain Basics.................................................................................. 3.3 Topology of Transition Matrix: Communication and Period.................. 3.4 The Perron-Frobenius Theorem.................................................................. 3.5 Convergence Measures............................................................................... 3.6 Markov Chains in Continuous or Heterogeneous State Spaces .............. 3.7 Ergodicity Theorem .................................................................................... 3.8 MCMC for Optimization by Simulated Annealing.................................. 3.8.1 Page Rank Example...................................................................... 3.9 Exercises...................................................................................................... References............................................................................................................... 49 50 50 53 56 58 61 62 62 65 67 70 4 Metropolis Methods andVariants....................................................................... 4.1 Introduction.................................................................................................. 4.2 The Metropolis-Hastings Algorithm.......................................................... 4.2.1 The Original Metropolis-
Hastings Algorithm........................... 4.2.2 Another Version of the MetropoUs-Hastings Algorithm........... 4.2.3 Other Acceptance Probability Designs........................................ 4.2.4 Key Issues in Metropolis Design................................................. 4.3 The Independence Metropolis Sampler..................................................... 4.3.1 The Eigenstructure of the IMS.................................................... 4.3.2 General First Hitting Time for Finite Spaces ............................ 4.3.3 Hitting Time Analysis for the IMS ............................................ 4.4 Reversible Jumps and Trans-Dimensional MCMC.................................. 4.4.1 Reversible Jumps........................................................................... 4.4.2 Toy Example: ID Range Image Segmentation.......................... 4.5 Application: Counting People..................................................................... 4.5.1 Marked Point Process Model....................................................... 4.5.2 Inference by MCMC..................................................................... 4.5.3 Results............................................................................................ 4.6 Application: Furniture Arrangement ........................................................ 4.7 Application: Scene Synthesis..................................................................... 4.8 Exercises......................................................................................................
References............................................................................................................... 71 71 72 72 74 74 75 75 77 78 78 80 80 81 85 85 86 87 88 90 95 96 5 Gibbs Sampler and Its Variants........................................................................... 5.1 Introduction.................................................................................................. 5.2 Gibbs Sampler............................................................................................. 5.2.1 A Major Problem with the Gibbs Sampler................................... 97 97 99 101
Contents IX 5.3 Gibbs Sampler Generalizations.................................................................. 102 5.3.1 Hit-and-Run.................................................................................. 103 5.3.2 Generalized Gibbs Sampler ........................................................ 103 5.3.3 Generalized Hit-and-Run............................................................. 104 5.3.4 Sampling with Auxiliary Variables.............................................. 105 5.3.5 Simulated Tempering................................................................... 105 5.3.6 Slice Sampling.............................................................................. 106 5.3.7 Data Augmentation...................................................................... 107 5.3.8 Metropolized Gibbs Sampler....................................................... 108 5.4 Data Association and Data Augmentation ............................................... 110 5.5 Julesz Ensemble and MCMC Samplingof Texture................................... 112 5.5.1 The Julesz Ensemble: A Mathematical Definition of Texture....................................................................................... 112 5.5.2 The Gibbs Ensemble and Ensemble Equivalence..................... 115 5.5.3 Sampling the Julesz Ensemble.................................................... 116 5.5.4 Experiment: Sampling the Julesz Ensemble.............................. 117 5.6 Exercises...................................................................................................... 119
References............................................................................................................... 120 6 Cluster Sampling Methods.................................................................................. 123 6.1 Introduction.................................................................................................. 123 6.2 Potts Model and Swendsen-Wang ............................................................. 124 6.3 Interpretations of the SW Algorithm ........................................................ 127 6.3.1 Interpretation 1 : Metropolis-Hastings Perspective .................... 128 6.3.2 Interpretation 2: Data Augmentation........................................... 131 6.4 Some Theoretical Results........................................................................... 135 6.5 Swendsen-Wang Cuts for Arbitrary Probabilities ................................... 137 6.5.1 Step 1 : Data-Driven Clustering .................................................. 138 6.5.2 Step 2: Color Flipping................................................................. 139 6.5.3 Step 3: Accepting the Flip........................................................... 140 6.5.4 Complexity Analysis..................................................................... 142 6.6 Variants of the Cluster Sampling Method................................................. 142 6.6.1 Cluster Gibbs Sampling: The “Hit-and-Run” Perspective........ 143 6.6.2 The Multiple Flipping Scheme.................................................... 144 6.7 Application: Image
Segmentation ............................................................. 145 6.8 Multigrid and Multi-level SW-cut ............................................................. 148 6.8.1 SW-Cuts at Multigrid .................................................................. 150 6.8.2 SW-cuts at Multi-level ................................................................ 152 6.9 Subspace Clustering.................................................................................... 153 6.9.1 Subspace Clustering by Swendsen-Wang Cuts........................... 155 6.9.2 Application: Sparse Motion Segmentation ................................ 158
x Contents 6.10 C4: Clustering Cooperative and Competitive Constraints....................... 163 6.10.1 Overview of the C4 Algorithm..................................................... 165 6.10.2 Graphs, Coupling, and Clustering .............................................. 166 6.10.3 C4 Algorithm on Flat Graphs ...................................................... 172 6.10.4 Experiments on Flat Graphs ....................................................... 175 6.10.5 Checkerboard Ising Model .......................................................... 176 6.10.6 C4 on Hierarchical Graphs ......................................................... 181 6.10.7 Experiments on Hierarchical C4................................................. 183 6.11 Exercises...................................................................................................... 184 References............................................................................................................... 186 7 Convergence Analysis of MCMC........................................................................ 7.1 Introduction.................................................................................................. 7.2 Key Convergence Topics............................................................................ 7.3 Practical Methods for Monitoring............................................................. 7.4 Coupling Methods for Card Shuffling....................................................... 7.4.1 Shuffling to the
Top...................................................................... 7.4.2 Riffle Shuffling.............................................................................. 7.5 Geometric Bounds, Bottleneck and Conductance..................................... 7.5.1 Geometric Convergence............................................................... 7.6 Peskun’s Ordering and Ergodicity Theorem.............................................. 7.7 Path Coupling and Exact Sampling............................................................ 7.7.1 Coupling From the Past................................................................ 7.7.2 Application: Sampling the Ising Model...................................... 7.8 Exercises...................................................................................................... References............................................................................................................... 189 189 189 191 193 193 194 196 196 200 201 202 203 205 209 8 Data Driven Markov Chain Monte Carlo ........................................................ 8.1 Introduction.................................................................................................. 8.2 Issues with Segmentation and Introduction to DDMCMC...................... 8.3 Simple Illustration of the DDMCMC........................................................ 8.3.1 Designing MCMC: The Basic Issues......................................... 8.3.2 Computing Proposal Probabilities in the Atomic Spaces: Atomic
Particles............................................................................ 8.3.3 Computing Proposal Probabilities in Object Spaces: Object Particles ............................................................................ 8.3.4 Computing Multiple, Distinct Solutions: Scene Particles......... 8.3.5 The Ψ-World Experiment............................................................ 8.4 Problem Formulation and Image Models ................................................. 8.4.1 Bayesian Formulation for Segmentation ................................... 8.4.2 The Prior Probability ................................................................... 8.4.3 The Likelihood for Grey Level Images....................................... 211 211 211 213 216 217 219 220 221 223 223 224 224
Contents 9 XI 8.4.4 Model Calibration ........................................................................ 8.4.5 Image Models for Color.............................................................. 8.5 Anatomy of Solution Space ...................................................................... 8.6 Exploring the Solution Space by Ergodic Markov Chains ..................... 8.6.1 Five Markov Chain Dynamics..................................................... 8.6.2 The Bottlenecks ........................................................................... 8.7 Data-Driven Methods.................................................................................. 8.7.1 Method I: Clustering in Atomic Spaces .................................... 8.7.2 Method II: Edge Detection.......................................................... 8.8 Computing Importance Proposal Probabilities ........................................ 8.9 Computing Multiple Distinct Solutions .................................................... 8.9.1 Motivation and a Mathematical Principle.................................... 8.9.2 A К-Adventurers Algorithm for Multiple Solutions ............... 8.10 Image Segmentation Experiments............................................................. 8.11 Application: Image Parsing........................................................................ 8.11.1 Bottom-Up and Top-Down Processing........................................ 8.11.2 Generative and Discriminative Methods..................................... 8.11.3 Markov Chain Kernels and Sub-Kernels
................................... 8.11.4 DDMCMC and Proposal Probabilities ....................................... 8.11.5 The Markov Chain Kernels........................................................... 8.11.6 Image Parsing Experiments ........................................................ 8.12 Exercises...................................................................................................... References............................................................................................................... 226 228 229 229 230 232 233 233 237 237 242 242 243 244 247 250 251 252 254 264 272 276 277 Hamiltonian and Langevin Monte Carlo.......................................................... 9.1 Introduction.................................................................................................. 9.2 Hamiltonian Mechanics.............................................................................. 9.2.1 Hamilton’s Equations................................................................... 9.2.2 A Simple Model of HMC............................................................. 9.3 Properties of Hamiltonian Mechanics....................................................... 9.3.1 Conservation of Energy................................................................ 9.3.2 Reversibility.................................................................................. 9.3.3 Symplectic Structure and Volume Preservation........................ 9.4 The Leapfrog Discretization of Hamilton’s Equations............................. 9.4.1 Euler’s
Method.............................................................................. 9.4.2 Modified Euler’s Method............................................................. 9.4.3 The Leapfrog Integrator.............................................................. 9.4.4 Properties of the Leapfrog Integrator......................................... 9.5 Hamiltonian Monte Carlo and Langevin Monte Carlo............................. 9.5.1 Formulation of HMC ................................................................... 9.5.2 The HMC Algorithm..................................................................... 9.5.3 The LMC Algorithm..................................................................... 281 281 282 282 283 284 285 286 286 288 288 288 289 290 292 292 293 295
xii Contents 9.5.4 Tuning НМС................................................................................... 9.5.5 Proof of Detailed Balance for HMC............................................. 9.6 Riemann Manifold HMC.............................................................................. 9.6.1 Linear Transformations in HMC................................................... 9.6.2 RMHMC Dynamics........................................................................ 9.6.3 RMHMC Algorithm and Variants................................................. 9.6.4 Covariance Functions in RMHMC................................................ 9.7 HMC in Practice............................................................................................ 9.7.1 Simulated Experiments on Constrained Normal Distributions.................................................................................... 9.7.2 Sampling Logistic Regression Coefficients withRMHMC....... 9.7.3 Sampling Image Densities with LMC: FRAME, GRADE and DeepFRAME........................................................................... 9.8 Exercises........................................................................................................ References................................................................................................................. 10 11 298 299 301 301 305 307 308 310 310 313 318 323 325 Learning with Stochastic Gradient ..................................................................... 10.1
Introduction................................................................................................... 10.2 Stochastic Gradient: Motivation and Properties.......................................... 10.2.1 Motivating Cases............................................................................. 10.2.2 Robbins-Monro Theorem............................................................... 10.2.3 Stochastic Gradient Descent and the Langevin Equation......... 10.3 Parameter Estimation for Markov Random Field (MRF) Models........... 10.3.1 Learning a FRAME Model with Stochastic Gradient................. 10.3.2 Alternate Methods of Learning for FRAME................................ 10.3.3 Four Variants of the FRAME Algorithm....................................... 10.3.4 Experiments ................................................................................... 10.4 Learning Image Models with Neural Networks.......................................... 10.4.1 Contrastive Divergence and Persistent Contrastive Divergence ...................................................................................... 10.4.2 Learning a Potential Energy for Images with Deep Networks: DeepFRAME................................................................ 10.4.3 Generator Networks and Alternating Backward Propagation... 10.4.4 Cooperative Energy and Generator Models.................................. 10.5 Exercises........................................................................................................
References................................................................................................................. 327 327 327 328 331 333 336 337 339 341 345 349 351 355 360 364 365 Mapping the EnergyLandscape ........................................................................... 11.1 Introduction.................................................................................................... 11.2 Landscape Examples, Structures, and Tasks ............................................. 11.2.1 Energy-Based Partitions of the State Space.................................. 11.2.2 Constructing a Disconnectivity Graph.......................................... 367 367 367 371 372 350
Contents xiii 11.2.3 11.2.4 ELM Example in 2D.................................................................... Characterizing the Difficulty (or Complexity) of Learning Tasks ........................................................................ 11.3 Generalized Wang-Landau Algorithm....................................................... 11.3.1 Barrier Estimation for GWL Mapping....................................... 11.3.2 Volume Estimation with GWL.................................................... 11.3.3 GWL Convergence Analysis....................................................... 11.4 GWL Experiments....................................................................................... 11.4.1 GWL Mappings of Gaussian Mixture Models........................... 11.4.2 GWL Mapping of Grammar Models........................................... 11.5 Mapping the Landscape with Attraction-Diffusion.................................. 11.5.1 Metastability and a Macroscopic Partition................................. 11.5.2 Introduction to Attraction-Diffusion........................................... 11.5.3 Attraction-Diffusion and the Ising Model ................................. 11.5.4 Attraction-Diffusion ELM Algorithm......................................... 11.5.5 Tuning ADELM............................................................................ 11.5.6 Barrier Estimation with AD........................................................ 11.6 Mapping the SK Spin Glass Model with GWLand ADELM.................. 11.7 Mapping Image Spaces with Attraction-
Diffusion................................... 11.7.1 Structure of Image Galaxies........................................................ 11.7.2 Experiments.................................................................................. 11.8 Exercises...................................................................................................... References............................................................................................................... Index 374 375 377 379 380 382 383 383 389 396 396 398 400 402 405 405 407 410 410 413 417 418 421
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| any_adam_object | 1 |
| author | Barbu, Adrian G. 1971- Zhu, Song-Chun |
| author_GND | (DE-588)1206761792 (DE-588)1206762624 |
| author_facet | Barbu, Adrian G. 1971- Zhu, Song-Chun |
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| bvnumber | BV046576707 |
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| discipline | Mathematik |
| format | Book |
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| id | DE-604.BV046576707 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-10T08:47:46Z |
| institution | BVB |
| institution_GND | (DE-588)1065365012 |
| isbn | 9789811329708 9811329702 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-031988562 |
| oclc_num | 1145243584 |
| open_access_boolean | |
| owner | DE-739 DE-706 DE-188 |
| owner_facet | DE-739 DE-706 DE-188 |
| physical | xvi, 422 Seiten Ilustrationen; Diagramme 23.5 cm x 15.5 cm |
| publishDate | 2020 |
| publishDateSearch | 2020 |
| publishDateSort | 2020 |
| publisher | Springer |
| record_format | marc |
| spelling | Barbu, Adrian G. 1971- Verfasser (DE-588)1206761792 aut Monte Carlo methods Adrian Barbu ; Song-Chun Zhu Springer [2020] © 2020 xvi, 422 Seiten Ilustrationen; Diagramme 23.5 cm x 15.5 cm txt rdacontent n rdamedia nc rdacarrier Monte-Carlo-Simulation (DE-588)4240945-7 gnd rswk-swf Hardback COM077000 UYAM COM016000 MAT029000 UYQV PBT B SCI17036: Probability and Statistics in Computer Science SUCO11645: Computer Science SCI22005: Computer Imaging, Vision, Pattern Recognition and Graphics SCS11001: Statistical Theory and Methods SCS17020: Statistics for Engineering, Physics, Computer Science, Chemistry and Earth Sciences 1627: Hardcover, Softcover / Mathematik/Wahrscheinlichkeitstheorie, Stochastik, Mathematische Statistik Monte-Carlo-Simulation (DE-588)4240945-7 s DE-604 Zhu, Song-Chun Verfasser (DE-588)1206762624 aut Springer Malaysia Representative Office (DE-588)1065365012 pbl Erscheint auch als Online-Ausgabe 9789811329715 X:MVB text/html http://deposit.dnb.de/cgi-bin/dokserv?id=f97b3cf2d6544b3eb997b1ead8aa72bd&prov=M&dok_var=1&dok_ext=htm Inhaltstext X:MVB http://www.springer.com/ Digitalisierung UB Passau - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=031988562&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Barbu, Adrian G. 1971- Zhu, Song-Chun Monte Carlo methods Monte-Carlo-Simulation (DE-588)4240945-7 gnd |
| subject_GND | (DE-588)4240945-7 |
| title | Monte Carlo methods |
| title_auth | Monte Carlo methods |
| title_exact_search | Monte Carlo methods |
| title_full | Monte Carlo methods Adrian Barbu ; Song-Chun Zhu |
| title_fullStr | Monte Carlo methods Adrian Barbu ; Song-Chun Zhu |
| title_full_unstemmed | Monte Carlo methods Adrian Barbu ; Song-Chun Zhu |
| title_short | Monte Carlo methods |
| title_sort | monte carlo methods |
| topic | Monte-Carlo-Simulation (DE-588)4240945-7 gnd |
| topic_facet | Monte-Carlo-Simulation |
| url | http://deposit.dnb.de/cgi-bin/dokserv?id=f97b3cf2d6544b3eb997b1ead8aa72bd&prov=M&dok_var=1&dok_ext=htm http://www.springer.com/ http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=031988562&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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