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DD2434 - Advanced Machine Learning Gaussian Processes Carl Henrik - PowerPoint PPT Presentation

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Introduction Recap Kernels Gaussian Processes References DD2434 - Advanced Machine Learning Gaussian Processes Carl Henrik Ek { chek } @csc.kth.se Royal Institute of Technology November 5th, 2015 Ek KTH DD2434 - Advanced Machine Learning

  1. Introduction Recap Kernels Gaussian Processes References Kernels Kernels and covariances • Covariance between columns: X T Y (data-dimensions) • Covariance between rows: XY T (data-points) • Kernels: k ( x , y ) = ϕ ( x ) T ϕ ( y ) ▶ Kernel functions are covariances between data-points • A kernel function describes the co-variance of the data points • Specific class of functions Ek KTH DD2434 - Advanced Machine Learning

  2. Introduction Recap Kernels Gaussian Processes References Kernels Kernels and covariances • Covariance between columns: X T Y (data-dimensions) • Covariance between rows: XY T (data-points) • Kernels: k ( x , y ) = ϕ ( x ) T ϕ ( y ) ▶ Kernel functions are covariances between data-points • A kernel function describes the co-variance of the data points • Specific class of functions Ek KTH DD2434 - Advanced Machine Learning

  3. Introduction Recap Kernels Gaussian Processes References Kernels Kernels and covariances • Covariance between columns: X T Y (data-dimensions) • Covariance between rows: XY T (data-points) • Kernels: k ( x , y ) = ϕ ( x ) T ϕ ( y ) ▶ Kernel functions are covariances between data-points • A kernel function describes the co-variance of the data points • Specific class of functions Ek KTH DD2434 - Advanced Machine Learning

  4. Introduction Recap Kernels Gaussian Processes References Kernels Kernels and covariances • Covariance between columns: X T Y (data-dimensions) • Covariance between rows: XY T (data-points) • Kernels: k ( x , y ) = ϕ ( x ) T ϕ ( y ) ▶ Kernel functions are covariances between data-points • A kernel function describes the co-variance of the data points • Specific class of functions Ek KTH DD2434 - Advanced Machine Learning

  5. Introduction Recap Kernels Gaussian Processes References Kernels Kernels and covariances • Covariance between columns: X T Y (data-dimensions) • Covariance between rows: XY T (data-points) • Kernels: k ( x , y ) = ϕ ( x ) T ϕ ( y ) ▶ Kernel functions are covariances between data-points • A kernel function describes the co-variance of the data points • Specific class of functions Ek KTH DD2434 - Advanced Machine Learning

  6. Introduction Recap Kernels Gaussian Processes References Kernels Kernels and covariances • Covariance between columns: X T Y (data-dimensions) • Covariance between rows: XY T (data-points) • Kernels: k ( x , y ) = ϕ ( x ) T ϕ ( y ) ▶ Kernel functions are covariances between data-points • A kernel function describes the co-variance of the data points • Specific class of functions Ek KTH DD2434 - Advanced Machine Learning

  7. Introduction Recap Kernels Gaussian Processes References Kernels 2 ℓ 2 ( x i − x j ) T ( x i − x j ) 1 k ( x i , x j ) = σ 2 e − (17) Squared Exponential • How does the data vary along the dimensions spanned by the data • RBF, Squared Exponential, Exponentiated Quadratic • Co-variance smoothly decays with distance Ek KTH DD2434 - Advanced Machine Learning

  8. Introduction Recap Kernels Gaussian Processes References Building Kernels Expression Conditions k ( x , z ) = c k 1 ( x , z ) c - any non negative real constant. k ( x , z ) = f ( x ) k 1 ( x , z ) f ( z ) f - any real-valued function. k ( x , z ) = q ( k 1 ( x , z )) q - any polynomial with non-negative coefficients. k ( x , z ) = exp ( k 1 ( x , z )) k ( x , z ) = k 1 ( x , z ) + k 2 ( x , z ) k ( x , z ) = k 1 ( x , z ) k 2 ( x , z ) k ( x , z ) = k 3 ( φ ( x ) , φ ( z )) k 3 - valid kernel in the space mapped by φ . k ( x , z ) = h Ax , z i = h x , Az i A - symmetric psd matrix. k ( x , z ) = k a ( x a , z a ) + k b ( x b , z b ) x a and x b - non-necessarily disjoint partitions of x ; k ( x , z ) = k a ( x a , z a ) k b ( x b , z b ) k a and k b - valid kernels on their respective spaces. Ek KTH DD2434 - Advanced Machine Learning

  9. Introduction Recap Kernels Gaussian Processes References Summary • Defines inner products in some space • We don’t need to know the space, its implicitly defined by the kernel function • Defines co-variance between data-points Ek KTH DD2434 - Advanced Machine Learning

  10. Introduction Recap Kernels Gaussian Processes References Summary • Defines inner products in some space • We don’t need to know the space, its implicitly defined by the kernel function • Defines co-variance between data-points Ek KTH DD2434 - Advanced Machine Learning

  11. Introduction Recap Kernels Gaussian Processes References Summary • Defines inner products in some space • We don’t need to know the space, its implicitly defined by the kernel function • Defines co-variance between data-points Ek KTH DD2434 - Advanced Machine Learning

  12. Introduction Recap Kernels Gaussian Processes References Introduction Recap Kernels Gaussian Processes Ek KTH DD2434 - Advanced Machine Learning

  13. Introduction Recap Kernels Gaussian Processes References What have you seen up till now? • Probabilistic modelling ▶ likelihood, prior, posterior ▶ marginalisation • Implicit feature spaces ▶ kernel functions • We have assumed the form of the mapping without uncertainty Ek KTH DD2434 - Advanced Machine Learning

  14. Introduction Recap Kernels Gaussian Processes References What have you seen up till now? • Probabilistic modelling ▶ likelihood, prior, posterior ▶ marginalisation • Implicit feature spaces ▶ kernel functions • We have assumed the form of the mapping without uncertainty Ek KTH DD2434 - Advanced Machine Learning

  15. Introduction Recap Kernels Gaussian Processes References Outline • General Regression • Introduce uncertainty in mapping • prior over the space of functions Ek KTH DD2434 - Advanced Machine Learning

  16. Introduction Recap Kernels Gaussian Processes References Outline • General Regression • Introduce uncertainty in mapping • prior over the space of functions Ek KTH DD2434 - Advanced Machine Learning

  17. Introduction Recap Kernels Gaussian Processes References Outline • General Regression • Introduce uncertainty in mapping • prior over the space of functions Ek KTH DD2434 - Advanced Machine Learning

  18. Introduction Recap Kernels Gaussian Processes References Regression Regression model, y i = f ( x i ) + ϵ (18) ϵ ∼ N ( 0 , σ 2 I ) (19) Introduce f i as instansiation of function, f i = f ( x i ) , (20) as a new random variable. Ek KTH DD2434 - Advanced Machine Learning

  19. Introduction Recap Kernels Gaussian Processes References Regression Model, p ( Y , f , X , θ ) = p ( Y | f ) p ( f | X , θ ) p ( X ) p ( θ ) (21) Want to “push” X through a mapping f of which we are uncertain, p ( f | X , θ ) , (22) prior over instansiations of function. Ek KTH DD2434 - Advanced Machine Learning

  20. Introduction Recap Kernels Gaussian Processes References Priors over functions 3 3 Lecture7/gp basics.py Ek KTH DD2434 - Advanced Machine Learning

  21. Introduction Recap Kernels Gaussian Processes References Priors over functions 3 3 Lecture7/gp basics.py Ek KTH DD2434 - Advanced Machine Learning

  22. Introduction Recap Kernels Gaussian Processes References Priors over functions 3 3 Lecture7/gp basics.py Ek KTH DD2434 - Advanced Machine Learning

  23. Introduction Recap Kernels Gaussian Processes References Priors over functions 3 3 Lecture7/gp basics.py Ek KTH DD2434 - Advanced Machine Learning

  24. Introduction Recap Kernels Gaussian Processes References Priors over functions 3 3 Lecture7/gp basics.py Ek KTH DD2434 - Advanced Machine Learning

  25. Introduction Recap Kernels Gaussian Processes References Gaussian Distribution Joint Distribution, [ x 1 ([ µ 1 [ σ ( x 1 , x 1 ) ] ] ]) σ ( x 1 , x 2 ) ∼ N , . (23) x 2 µ 2 σ ( x 2 , x 1 ) σ ( x 2 , x 2 ) µ 2 + σ ( x 1 , x 2 ) σ ( x 1 , x 1 ) − 1 ( x 1 − µ 1 ) , x 2 | x 1 ∼ N ( σ ( x 2 , x 2 ) − σ ( x 2 , x 1 ) σ ( x 1 , x 1 ) − 1 σ ( x 1 , x 2 ) ) (24) Ek KTH DD2434 - Advanced Machine Learning

  26. Introduction Recap Kernels Gaussian Processes References The Gaussian Conditional 4 ([ 0 ] [ ]) 1 0 . 5 N , (25) 0 0 . 5 1 4 Lecture7/conditional gaussian.py Ek KTH DD2434 - Advanced Machine Learning

  27. Introduction Recap Kernels Gaussian Processes References The Gaussian Conditional 4 ([ 0 ] [ ]) 1 0 . 5 N , (26) 0 0 . 5 1 4 Lecture7/conditional gaussian.py Ek KTH DD2434 - Advanced Machine Learning

  28. Introduction Recap Kernels Gaussian Processes References The Gaussian Conditional 4 ([ 0 ] [ ]) 1 0 . 5 N , (27) 0 0 . 5 1 4 Lecture7/conditional gaussian.py Ek KTH DD2434 - Advanced Machine Learning

  29. Introduction Recap Kernels Gaussian Processes References The Gaussian Conditional 4 ([ 0 ] [ ]) 1 0 . 5 N , (28) 0 0 . 5 1 4 Lecture7/conditional gaussian.py Ek KTH DD2434 - Advanced Machine Learning

  30. Introduction Recap Kernels Gaussian Processes References The Gaussian Conditional 4 ([ 0 ] [ ]) 1 0 . 5 N , (29) 0 0 . 5 1 4 Lecture7/conditional gaussian.py Ek KTH DD2434 - Advanced Machine Learning

  31. Introduction Recap Kernels Gaussian Processes References The Gaussian Conditional 4 ([ 0 ] [ ]) 1 0 . 99 N , (30) 0 0 . 99 1 4 Lecture7/conditional gaussian.py Ek KTH DD2434 - Advanced Machine Learning

  32. Introduction Recap Kernels Gaussian Processes References The Gaussian Conditional 4 ([ 0 ] [ ]) 1 0 . 99 N , (31) 0 0 . 99 1 4 Lecture7/conditional gaussian.py Ek KTH DD2434 - Advanced Machine Learning

  33. Introduction Recap Kernels Gaussian Processes References The Gaussian Conditional 4 ([ 0 ] [ ]) 1 0 . 99 N , (32) 0 0 . 99 1 4 Lecture7/conditional gaussian.py Ek KTH DD2434 - Advanced Machine Learning

  34. Introduction Recap Kernels Gaussian Processes References The Gaussian Conditional 4 ([ 0 ] [ ]) 1 0 . 99 N , (33) 0 0 . 99 1 4 Lecture7/conditional gaussian.py Ek KTH DD2434 - Advanced Machine Learning

  35. Introduction Recap Kernels Gaussian Processes References The Gaussian Conditional 4 ([ 0 ] [ ]) 1 0 . 99 N , (34) 0 0 . 99 1 4 Lecture7/conditional gaussian.py Ek KTH DD2434 - Advanced Machine Learning

  36. Introduction Recap Kernels Gaussian Processes References The Gaussian Conditional 4 ([ 0 [ 1 ] ]) 0 N , (35) 0 0 1 4 Lecture7/conditional gaussian.py Ek KTH DD2434 - Advanced Machine Learning

  37. Introduction Recap Kernels Gaussian Processes References The Gaussian Conditional 4 ([ 0 [ 1 ] ]) 0 N , (36) 0 0 1 4 Lecture7/conditional gaussian.py Ek KTH DD2434 - Advanced Machine Learning

  38. Introduction Recap Kernels Gaussian Processes References The Gaussian Conditional 4 ([ 0 [ 1 ] ]) 0 N , (37) 0 0 1 4 Lecture7/conditional gaussian.py Ek KTH DD2434 - Advanced Machine Learning

  39. Introduction Recap Kernels Gaussian Processes References The Gaussian Conditional 4 ([ 0 [ 1 ] ]) 0 N , (38) 0 0 1 4 Lecture7/conditional gaussian.py Ek KTH DD2434 - Advanced Machine Learning

  40. Introduction Recap Kernels Gaussian Processes References The Gaussian Conditional 4 ([ 0 [ 1 ] ]) 0 N , (39) 0 0 1 4 Lecture7/conditional gaussian.py Ek KTH DD2434 - Advanced Machine Learning

  41. Introduction Recap Kernels Gaussian Processes References Ek KTH DD2434 - Advanced Machine Learning

  42. Introduction Recap Kernels Gaussian Processes References Ek KTH DD2434 - Advanced Machine Learning

  43. Introduction Recap Kernels Gaussian Processes References Ek KTH DD2434 - Advanced Machine Learning

  44. Introduction Recap Kernels Gaussian Processes References Ek KTH DD2434 - Advanced Machine Learning

  45. Introduction Recap Kernels Gaussian Processes References Ek KTH DD2434 - Advanced Machine Learning

  46. Introduction Recap Kernels Gaussian Processes References Ek KTH DD2434 - Advanced Machine Learning

  47. Introduction Recap Kernels Gaussian Processes References Ek KTH DD2434 - Advanced Machine Learning

  48. Introduction Recap Kernels Gaussian Processes References Ek KTH DD2434 - Advanced Machine Learning

  49. Introduction Recap Kernels Gaussian Processes References Ek KTH DD2434 - Advanced Machine Learning

  50. Introduction Recap Kernels Gaussian Processes References If all instansiations of the function is jointly Gaussian such that the co-variance structure depends on how much information an observation provides for the other we will get the curve above. Ek KTH DD2434 - Advanced Machine Learning

  51. Introduction Recap Kernels Gaussian Processes References Row space • Co-variance between each point! • Co-variance function is a kernel! • We can do all this in induced space, i.e. allow for any function! Ek KTH DD2434 - Advanced Machine Learning

  52. Introduction Recap Kernels Gaussian Processes References Row space • Co-variance between each point! • Co-variance function is a kernel! • We can do all this in induced space, i.e. allow for any function! Ek KTH DD2434 - Advanced Machine Learning

  53. Introduction Recap Kernels Gaussian Processes References Row space • Co-variance between each point! • Co-variance function is a kernel! • We can do all this in induced space, i.e. allow for any function! Ek KTH DD2434 - Advanced Machine Learning

  54. Introduction Recap Kernels Gaussian Processes References Gaussian Processes 5 p ( f | X , θ ) ∼ GP ( µ ( X ) , k ( X , X )) (40) Defenition A Gaussian Process is an infinite collection of random variables who any subset is jointly gaussian. The process is specified by a mean function µ ( · ) and a co-variance function k ( · , · ) f ∼ GP ( µ ( · ) , k ( · , · )) (41) 5 Bishop 2006, p. 6.4.2 Ek KTH DD2434 - Advanced Machine Learning

  55. Introduction Recap Kernels Gaussian Processes References Gaussian Processes 5 p ( f | X , θ ) ∼ GP ( µ ( X ) , k ( X , X )) (42) y i = f i + ϵ (43) ϵ ∼ N ( 0 , σ 2 I ) (44) ∫ p ( Y | X , θ ) = p ( Y | f ) p ( f | X , θ ) d f (45) Connection to Distribution GP is infinite, but we only observe finite amount of data. This means conditioning on a subset of the data, the GP is a just a Gaussian distribution, which is self-conjugate. 5 Bishop 2006, p. 6.4.2 Ek KTH DD2434 - Advanced Machine Learning

  56. Introduction Recap Kernels Gaussian Processes References Gaussian Processes 5 The mean function • Function of only the input location • What do I expect the function value to be only accounting for the input location • We will assume this to be constant The co-variance function • Function of two input locations • How should the information from other locations with known function value observations effect my estimate • Encodes the behavior of the function 5 Bishop 2006, p. 6.4.2 Ek KTH DD2434 - Advanced Machine Learning

  57. Introduction Recap Kernels Gaussian Processes References Gaussian Processes 5 The mean function • Function of only the input location • What do I expect the function value to be only accounting for the input location • We will assume this to be constant The co-variance function • Function of two input locations • How should the information from other locations with known function value observations effect my estimate • Encodes the behavior of the function 5 Bishop 2006, p. 6.4.2 Ek KTH DD2434 - Advanced Machine Learning

  58. Introduction Recap Kernels Gaussian Processes References Gaussian Processes 5 The mean function • Function of only the input location • What do I expect the function value to be only accounting for the input location • We will assume this to be constant The co-variance function • Function of two input locations • How should the information from other locations with known function value observations effect my estimate • Encodes the behavior of the function 5 Bishop 2006, p. 6.4.2 Ek KTH DD2434 - Advanced Machine Learning

  59. Introduction Recap Kernels Gaussian Processes References Gaussian Processes 5 The Prior p ( f | X , θ ) = GP ( µ ( x ) , k ( x , x ′ )) (46) µ ( x ) = 0 (47) 2 ℓ 2 ( x i − x j ) T ( x i − x j ) 1 k ( x i , x j ) = σ 2 e − (48) 5 Bishop 2006, p. 6.4.2 Ek KTH DD2434 - Advanced Machine Learning

  60. Introduction Recap Kernels Gaussian Processes References Gaussian Processes 5 5 Bishop 2006, p. 6.4.2 Ek KTH DD2434 - Advanced Machine Learning

  61. Introduction Recap Kernels Gaussian Processes References Gaussian Processes 5 5 Bishop 2006, p. 6.4.2 Ek KTH DD2434 - Advanced Machine Learning

  62. Introduction Recap Kernels Gaussian Processes References Gaussian Processes 5 5 Bishop 2006, p. 6.4.2 Ek KTH DD2434 - Advanced Machine Learning

  63. Introduction Recap Kernels Gaussian Processes References Gaussian Processes 5 5 Bishop 2006, p. 6.4.2 Ek KTH DD2434 - Advanced Machine Learning

  64. Introduction Recap Kernels Gaussian Processes References Gaussian Processes 5 5 Bishop 2006, p. 6.4.2 Ek KTH DD2434 - Advanced Machine Learning

  65. Introduction Recap Kernels Gaussian Processes References Gaussian Processes 5 5 Bishop 2006, p. 6.4.2 Ek KTH DD2434 - Advanced Machine Learning

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