In honor of Professor Amari Information geometry associated with - PowerPoint PPT Presentation

Information Geometry and its Applications IV June 12-17, 2016, Liblice, Czech In honor of Professor Amari Information geometry associated with two generalized means Shinto Eguchi Institute of Statistical Mathematics A joint work with Osamu Komori and Atsumi Ohara University of Fukui

Outline Information geometry Generalized information geometry 2

The core of information geometry Pythagoras (Amari-Nagaoka, 2001) 3

Metric and connections Information metric m-connection e-connection Conjugate Dawid (1975), Amari (1982) Rao (1945), 4

Exponential model Exponential model Mean parameter Remark 1 Amari (1982) Remark 2 5

Minimum KL leaf Exponential model Mean equal space 6

Pythagoras foliation 7

KL divergence (revisited) The normalizing constant We observe Cf. Ay-Amari (2015) KL divergence is induced by e-geodesic Cf . the canonical divergence , Amari-Nagaoka (2001) 8

(log , exp) (log, exp) KL-divergence e-geodesic m-geodesic Pythagoras identity exponential model Pythagoras foliation mean equal space 9

( log, exp) (  ,  ) ￥ 10

Kolmogorov-Nagumo mean Cf. Kolmogorov(1930), Nagumo (1930), Naudts (2009) 11

Generalized e-geodesic Remark 12

13 Generalized KL-divergence

14 Generalized m-geodesic Remark

Metric and connections by D (  ) Riemannian metric Generalized m-connection Generalized e-connection Remark 15

Generalized two geodesic curves 16

 Pythagorean theorem Pythagoras theorem 17

Power-log function 18

Generalized exponential model G-exponential model Mean parameter Cf. Naudts (2010) Remark 1 Cf. Matsuzoe-Henmi (2014) Remark 2 19

Minimum GKL leaf G-exponential model Mean equal space 20

Generalized Pythagoras foliation 21

( log , exp ) (  Cf. Newton (2012) 22

Expected  density Consider Let Then Therefore 23

Quasi divergence Remark 24

Another generalization of KL-divergence One adjustment The other adjustment 25

Gamma divergence Remark Fujisawa-Eguchi (2008) 26

    -geometry m-geodesic G-e-geodesic Pythagoras 27 27

Metric and connections by  (  ) Riemannian metric Generalized m-connection Generalized e-connection Remark 28

Minimum     estimation Model Data set Loss function Proposed estimator Expected loss consistency Estimation selection, Robustness, Spontaneous data learning 29 29

Non-convex learning Model Loss function Selection of  30

Spontaneous data learning The consistent class of estimators Super robustness Redescending Influence function Multi modal detection Difference of convex functions two local minima Estimation selection than model selection 31 31

Concluding remarks (  32

 Welcome to any comments: eguchi@ism.ac.jp 33

U -divergence U- cross-entropy U- entropy U- divergence Note Exm 34

Generalized exponential model (ver. 2) G-exponential model Mean parameter Remark 1 Remark 2 35