1. Dimension Reduction Based on
the Geometry of Dually Flat Spaces

2. • 2002.04 – 2006.03
• 2006.04 – 2012.03
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3. 1.
2.
3.
4.
5.
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4. 1.
2.
3.
4.
5.
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5. •
¾
–
–
–
9
–
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6. •
¾
–
–
•
¾
–
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7. ‡
–
•
–
–
→
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8. ‡
¾
– ridge LASSO
• LASSO L1
–
• LARS (Least Angle Regression)
–
– LASSO
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9. LARS
• LARS (Least Angle Regression)
– Efron, Hastie, Johnstone and Tibshirani: Least Angle
Regression (with discussion). Annals of Statistics,
vol. 32 (2004), pp. 407-499.
–
–
–
–
9
9
‹ LASSO
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10. 24 5 28 10

11. 24 5 28 11

12. 24 5 28 12

13. .
→ ≥
.
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14. LARS
[Efron, et al. (2004)]
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15. LASSO
[Efron, et al. (2004)]
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16. Stagewise
[Efron, et al. (2004)]
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17. LARS
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18. ‡
–
–
–
•
–
–
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19. •
•
•
– e- m-
• e- e-
–
• m- m-
–
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20. 24 5 28 20

21. 24 5 28 21

22. • LARS
–
•
• LARS LASSO
–
LARS LASSO
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23. 1.
2.
3.
4.
5.
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24. • Bisector Regression
Least Angle Regression
• BRGLM
– Bisector Regression for Generalized Linear Models
• BRGGM
– Bisector Regression for Gaussian Graphical Models
• BRCT ←
– Bisector Regression for Contingency Tables
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25. 24 5 28 25

26. 24 5 28 26

27. .
→ ≥
.
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28. •
24 5 28 28

29. •
•
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30. •
•
–
–
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31. 1.
2.
3.
4.
5.
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32. „
•
• Bisector regression
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33. •
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34. • e-
• m-
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35. • covariance matrix
⊧ ∽ ∨⊾≩≪ ∩ ∨⊧∩≩≪ ∽ ⊾≩≪
• Concentration matrix
⊧ ⊡∱ ∽ ∨⊾ ≩≪ ∩ ⊡⊧⊡∱⊢≩≪ ∽ ⊾≩≪
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36. •
•
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37. e-
m-
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38. Kullback-Leibler
•
• Kullback-Leibler
–
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39. m-
e- m-
m-
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40. Bisector Regression
•
–
–
1.
2.
3.
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41. Bisector Regression GGM
1.
2.
3.
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42. 24 5 28 42

43. 24 5 28 43

44. •
•
–
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45. • GGM ≢
⊧ ≩≪
⊩ ⊡∱ :
⊪
≍ ∨⊤∩ ∽ ⊧ ≪ ⊾ ∽ ∰ ∨∨≩∻ ≪ ∩ ∶∲ ⊤∩
≢ ≢
∨⊧∩≩≪ ∽ ∨⊧≍≌≅∩≩≪ ∨ ≩∻ ≪ ∩ ∲ ⊤
≢
∨⊧⊡∱∩≩≪ ∽ ∰ ∨≩∻ ≪ ∩ ∶∲ ⊤
≢
⊧≍≌≅
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46. BRGGM
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47. Bisector Regression
•
–
–
–
•
–
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48. Bisector Regression
•
–
• LASSO
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49. LARS
• LARS
• BR
→
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50. LARS
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51. BRGGM
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52. 24 5 28 52

53. 24 5 28 53

54. 1.
2.
3.
4.
5.
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55. •
• South Africa Heart Disease (SAHD) dataset
• R ElemStatLearn
•
–
•
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56. BRGLM Park and Hastie (2007)
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57. GLM
•
• Training, Test
– 100 10
• AIC
• 1000
BRGLM Park and
Hastie(2007)
0.30 0.30
(0.050) (0.054)
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58. •
•
• R SIN package
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59. 1
2 5
3 4
1
2 5
3 4
1: , 2: , 3: , 4: , 5:
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60. LASSOLin (2007)]
[Yuan and
1
2 5
3 4
1
2 5
3 4
1: , 2: , 3: , 4: , 5:
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61. • BRGGM graphical LASSO [Friedman, et al. (2007)]
– Graphical LASSO R glasso
• 1000 trials
– Training, test: 1000 observations
• AIC
•
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62. ≭≯≤≥≬ ≶≡≲≩≡≢≬≥ ≥≤≧≥ ≂≒≇≇≍ ≧≲≡≰≨≩≣≡≬ ≌≁≓≓≏
∨≩∩ ∵ ∳ ∱∮∰∱∱ ∨∰∮∰∲∳∩ ∱∮∰∶∵ ∨∰∮∰∲∴∩
∨≩≩∩ ∵ ∴ ∱∮∰∱∴ ∨∰∮∰∲∳∩ ∱∮∰∶∸ ∨∰∮∰∲∴∩
∨≩≩≩∩ ∶ ∷ ∱∮∲∲∸ ∨∰∮∰∲∶∩ ∱∮∲∹∱ ∨∰∮∰∲∷∩
∨≩≶∩ ∷ ∱∰ ∱∮∴∴∰ ∨∰∮∰∲∷∩ ∱∮∵∱∵ ∨∰∮∰∲∹∩
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63. • BRGGM
–
• 100 trials
– 100 observations
≤ ≡ ≴≡≳≥≴ ≶≡ ≲≩≡≢ ≬≥ ≥≤ ≧ ≥ ≴≲≵ ≥ ≭ ≯ ≤ ≥≬
∨≡∩ ∴ ∳ ∱∰∰∯∱∰∰
∨≢ ∩ ∵ ∵ ∹∹∯∱∰∰
∨≣∩ ∶ ∸ ∹∹∯∱∰∰
∨≤ ∩ ∱∰ ∲∲ ∱∲∯∱∰∰
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64. 1.
2.
3.
4.
5.
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65. •
–
•
–
•
¾
¾
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66. 24 5 28 66