Statistical Machine Learning Theory of Probabilistic Graphical Model

確率的グラフィカルモデルの統計的機械学習理論 Statistical Machine Learning Theory of Probabilistic Graphical Model 田中和之 東北大学大学院情報科学研究科 Kazuyuki Tanaka GSIS, Tohoku University, Sendai, Japan http: //www. smapip. is. tohoku. ac. jp/~kazu/ Collaborators Masayuki Ohzeki (Tohoku University, Japan) Muneki Yasuda (Yamagata University, Japan) Shun Kataoka (Otaru University of Commerce, Japan) Cyril Furtlehner (INRIA/Saclay, France) Candy Hsu (National Tsin Hua University, Taiwan) Mike Titterington (University of Glasgow, UK) June, 2018 Im. PACT school on Interdisciplinary Quantum Physics and Computer Science 1

Outline 1. 2. 3. 4. 5. 6. 7. 8. Introduction Statistical Machine Learning in Markov Random Fields Generalized Sparse Prior for Image Modeling Probabilistic Noise Reduction Probabilistic Image Segmentation Statistical Machine Learning by Inverse Renormalization Group Transformation Related Works Concluding Remarks June, 2018 Im. PACT school on Interdisciplinary Quantum Physics and Computer Science 3

Pairwise Markov Random Fields and Loopy Belief Propagation Probabilistic Information Processing Bayes Formulas Maximum Likelihood KL Divergence Markov Random Fields Probabilistic Models and Statistical Machine Learning Loopy Belief Propagation i j Message V: Set of all the nodes (vertices) in graph G E: Set of all the links (edges) in graph G June, 2018 Im. PACT school on Interdisciplinary Quantum Physics and Computer Science 4

Pairwise Markov Random Fields and Loopy Belief Propagation Message i 2 1 3 4 5 6 7 8 9 10 11 12 i i j j Message V: Set of all the nodes (vertices) in graph G E: Set of all the links (edges) in graph G June, 2018 Im. PACT school on Interdisciplinary Quantum Physics and Computer Science 5

Supervised Learning of Pairwise Markov Random Fields Pairwise MRF 1 2 3 4 5 6 7 8 9 10 11 12 Maximization of Likelihood V: Set of all the nodes (vertices) June, 2018 E: Set of all the links (edges) Im. PACT school on Interdisciplinary Quantum Physics and Computer Science 6

Pairwise Markov Random Fields for Bayesian Image Modeling In Bayesian image modeling, natural images are often assumed to be generated by according to the following pairwise Markov random fields: Hamiltonian of Classical Spin System June, 2018 Im. PACT school on Interdisciplinary Quantum Physics and Computer Science 7

Conventional Pairwise Markov Random Fields Gaussian Prior Huber Prior Saturated Quadratic Prior x June, 2018 x Convex Functions near x=0 Im. PACT school on Interdisciplinary Quantum Physics and Computer Science x 8

Supervised Learning of Pairwise Markov Random Fields Supervised Learning Scheme by Loopy Belief Propagation in Pairwise Markov random fields: Supervised Learning 30 Standard Images June, 2018 Im. PACT school on Interdisciplinary Quantum Physics and Computer Science 9

Supervised Learning of Pairwise Markov Random Fields by Loopy Belief Propagation Supervised Learning Scheme by Loopy Belief Propagation in Pairwise Markov random fields: Histogram from Supervised Data June, 2018 Im. PACT school on Interdisciplinary Quantum Physics and Computer Science M. Yasuda, S. Kataoka and K. Tanaka, J. Phys. Soc. Jpn, Vol. 81, No. 4, Article No. 044801, 2012. 10

Supervised Learning of Pairwise Markov Random Fields by Loopy Belief Propagation Supervised Learning Scheme by Loopy Belief Propagation in Pairwise Markov random fields: q=256 K. Tanaka, M. Yasuda and D. M. Titterington: J. Phys. Soc. Jpn, 81, 114802, 2012. June, 2018 Im. PACT school on Interdisciplinary Quantum Physics and Computer Science 11

Hyperparameter Estimation of Pairwise Markov Random Fields Data Generative Model Prior Probability Pairwise MRF Joint Probability Marginal Likelihood 5 9 2 3 4 5 6 7 8 9 10 11 12 2 1 Probability of Data d Likelihood of a and b 1 1 5 9 6 10 3 2 6 10 7 11 4 3 7 8 12 11 4 8 12 Maximization of Marginal Likelihood Expectation Maximization (EM) Algorithm V: Set of all the nodes (vertices) June, 2018 E: Set of all the links (edges) Im. PACT school on Interdisciplinary Quantum Physics and Computer Science 12

Degradation Process in Bayesian Image Modeling Assumption: Degraded image is generated from the original image by Additive White Gaussian Noise. June, 2018 Im. PACT school on Interdisciplinary Quantum Physics and Computer Science 13

Noise Reductions by Generalized Sparse Prior Deterministic Equations of a and b Repeat E step and M step until a(t) and s(t) converge June, 2018 Im. PACT school on Interdisciplinary Quantum Physics and Computer Science 14

Noise Reductions by Generalized Sparse Priors and Loopy Belief Propagation Original Image K. Tanaka, M. Yasuda and D. M. Titterington: J. Phys. Soc. Jpn, 81, 114802, 2012. Degraded Image K. Tanaka, M. Yasuda and D. M. Titterington: JPSJ, 81, 114802, 2012. Restored Image p=0. 3 June, 2018 p=2. 0 Im. PACT school on Interdisciplinary Quantum Physics and Computer Science Continuous Gaussian Graphical Model p=2. 0 15

Bayesian Modeling for Image Segmentation using MRF Data Parameter ai = 0, 1, …, q-1 Segment image data into some regions using belief propagation and EM algorithm Posterior Probability Dstribution Potts Prior 12 q+1 Hyperparameters Data Generative Model Likelihood of Hyperparameter Maximization of Posterior Marginal (MPM) Berkeley Segmentation Data Set 500 (BSDS 500), http: //www. eecs. berkeley. edu/Research/Projects/CS/vision/grouping/ P. Arbelaez, M. Maire, C. Fowlkes and J. Malik: IEEE Trans. PAMI, 33, 898, Im. PACT school on Interdisciplinary Quantum 2011. June, 2018 Physics and Computer Science 16

Bayesian Image Segmentation Intel® Core™ i 7 -4600 U CPU with a memory of 8 GB Potts Prior Hyperparameter q=8 321 x 481 Hyperparameter Estimation in Maximum Likelihood Framewrok and Maximization of Posterior Marginal (MPM) Estimation with Loopy Belief Propagation for Observed Color Image d K. Tanaka, S. Kataoka, M. Yasuda, Y. Waizumi and C. -T. Hsu: JPSJ, . 83, 124002, 2014. 481 x 321 June, 2018 Berkeley Segmentation Data Set 500 (BSDS 500), http: //www. eecs. berkeley. edu/Research/Projects/CS/vision/grouping/ P. Arbelaez, M. Maire, C. Fowlkes and J. Malik: IEEE Trans. PAMI, 33, 898, 2011. Im. PACT school on Interdisciplinary Quantum Physics and Computer Science 17

Bayesian Image Segmentation June, 2018 Im. PACT school on Interdisciplinary Quantum Physics and Computer Science 18

Real Space Renormalization Group Theory 1 Step 1 1 K K 2 K(1) K 1 K(1) 4 K K(1) 3 Step 2 June, 2018 3 2 K(1) 5 K K(1) 5 3 Im. PACT school on Interdisciplinary Quantum Physics and Computer Science K(1) 6 K 7 7 4 19

Real Space Renormalization Group Theory y y x x x If K(2) is given, the original value of K can be estimated by iterating June, 2018 Im. PACT school on Interdisciplinary Quantum Physics and Computer Science 20

Real Space. Renormalization Group Theory q=8 June, 2018 Im. PACT school on Interdisciplinary Quantum Physics and Computer Science 21

Bayesian Image Segmentation Potts Prior Hyperparameter Estimation in Maximization of Marginal Likelihood with Belief Propagation for Original Image Hyperparameter Estimation in Maximization of Marginal Likelihood with Belief Propagation after Coarse Graining Procedures Segmentation by Belief Propagation for Original Image 20 x 30 321 x 481 June, 2018 Coarse Graining Procedures (r =8) 20 x 30 Labeled Image Im. PACT school on Interdisciplinary Quantum Physics and Computer Science 22

Bayesian Image Segmentation Potts Prior 481 x 321 Hyperparameter Estimation in Maximum Likelihood Framework with Belief Propagation for Original Image q=8 Hyperparameter Estimation in Maximum Likelihood Framework with Belief Propagation after Coarse Graining Procedures Intel® Core™ i 7 -4600 U CPU with a memory of 8 GB Segmentation by Belief Propagation for Original Image MPM with LBP 30 x 20 Coarse Graining Labeled Image Procedures (r =8) Berkeley Segmentation Data Set 500 (BSDS 500), http: //www. eecs. berkeley. edu/Research/Projects/CS/vision/grouping/ June, 2018 Im. PACT school on Interdisciplinary Quantum Physics and Computer Science 23

Our related Works Image Inpainting Reconstruction of Traffic Density True data S. Kataoka, M. Yasuda and K. Tanaka: JPSJ, 81, 025001, 2012. June, 2018 reconstruction from 80% missing data S. Kataoka, M. Yasuda, C. Furtlehner and K. Tanaka: Inverse Problems, 30, 025003, 2014. Im. PACT school on Interdisciplinary Quantum Physics and Computer Science 24

Summary Supervised Learning of Pairwise Markov Random Fields Bayesian Image Modeling by Loopy Belief Propagation and EM algorithm. By introducing Inverse Real Space Renormalization Group Transformations, the computational time can be reduced. June, 2018 Im. PACT school on Interdisciplinary Quantum Physics and Computer Science 25

References 1. K. Tanaka, M. Yasuda and D. M. Titterington: Bayesian image 2. 3. 4. 5. modeling by means of generalized sparse prior and loopy belief propagation, Journal of the Physical Society of Japan, vol. 81, vo. 11, article no. 114802, November 2012. K. Tanaka, S. Kataoka, M. Yasuda, Y. Waizumi and C. -T. Hsu: Bayesian image segmentations by Potts prior and loopy belief propagation, Journal of the Physical Society of Japan, vol. 83, no. 12, article no. 124002, December 2014. K. Tanaka, S. Kataoka, M. Yasuda and M. Ohzeki: Inverse renormalization group transformation in Bayesian image segmentations, Journal of the Physical Society of Japan, vol. 84, no. 4, article no. 045001, April 2015. 田中和之著: ベイジアンネットワークの統計的推論の数理, コロナ社, October 2009. 田中和之 (分担執筆): 確率的グラフィカルモデル(鈴木譲, 植野真臣編), 第 8章 マルコフ確率場と確率的画像処理, pp. 195 -228, 共立出版, July 2016 June, 2018 Im. PACT school on Interdisciplinary Quantum Physics and Computer Science 26