1. Supervised and Unsupervised learning and application to Neuroscience Cours CA6b-4

2. Machine Learning2 A Generic System System … … Input Variables: Hidden Variables: Output Variables: Training examples: Parameters:

3. Machine Learning3 A Generic System System … … Input Variables: Hidden Variables: Output Variables: Training examples: Parameters:

4. Machine Learning4 Different types of learning Supervised learning: 1.Classification (discrete y), 2.Regression (continuous y). Unsupervised learning (no target y). 1.Clustering (h = different groups of types of data). 2.Density estimation (h = parameters of probability dist.) 3.Reduction (h= a few latent variable describing high dimensional data). Reinforcement learning (y = actions).

5. Digit recognition (supervised) Handwritten Digit Recognition x: pixelized or pre-processed image. t: classs of pre-classified digits (training example.) y: digit class (computed by ML algorithm). h: contours, left/right handed…

6. Regression (supervised) Target output Parameters

7. Linear classifier ? Training examples

8. Linear classifier Decision boundary Heavyside function: 0 1

9. Linear classifier Decision boundary Heavyside function: 0 1

10. Assumptions Multivariate Gaussians Same covariance Two classes equiprobable

11. How do we compute the output? Positive: Class 1 Negative: Class 0 Orthogonal to decision boundary

12. How do we compute the output? Orthogonal to decision boundary

13. How do we learn the parameters? Orthogonal to decision boundary Linear discriminant analysis = Direct parameter estimation

14. How do we learn the parameters? Orthogonal to decision boundary Minimize mean-squared error:

15. How do we learn the parameters? Minimize mean-squared error: Gradient descent:

16. How do we learn the parameters? Minimize mean-squared error: Gradient descent: Stochastic gradient descent:

17. How do we learn the parameters? Stochastic gradient descent: Problem:is not differentiable

18. 3. How do we learn the parameters? Solution: change y to expected class: The output is now the expected class Logistic function

19. 3. How do we learn the parameters? Stochastic gradient descent:

20. 3. How do we learn the parameters? Stochastic gradient descent: Always positive

21. 3. How do we learn the parameters? Learning based on expected class: with Perceptron learning rule with

22. Application 1: Neural population decoding

25. w

29. How to find ? w w

30. Linear Discriminant Analysis (LDA) Covariance Matrix: Mean responses:

31. Inverse Covariance matrix Average neural responses when motion is right Average neural responses when motion is left Linear Discriminant Analysis (LDA) w

32. Neural network interpretation: Learning the connections with « Delta rule »: Each neuron is a classifier

33. Limitation of 1 layer perceptron: Linearly separable: ANDNon linearly separable: XOR 0 1 1 0 1 1

34. Extension: multilayer perceptron Towards a universal computer 0 1 1 0 1 1

35. Learning a multi-layer neural network with backprop Towards a universal computer

36. Extension: multilayer perceptron Towards a universal computer Initial error:

37. Extension: multilayer perceptron Towards a universal computer Backpropagate errors Initial error:

38. Extension: multilayer perceptron Towards a universal computer Backpropagate errors Apply delta rule: Initial error:

39. Big problem: overfitting... … Backprop was abandoned in the late eighties…

40. Compensate with very large datasets 9 th Order Polynomial … Resurgence of backprop with big data

41. Deep convulational networks Google: Image recognition, speech recognition. Trained on billions of examples…

42. Single neurons as 2 layer perceptron Poirazi and Mel, 2001, 2003

43. Regression (supervised) Target output Parameters

44. Regression in general Target output Basis functions

45. Gaussian noise assumption

46. How to learn the parameters? Gradient descent:

47. But: overfitting...

48. How to learn the parameters? Gradient descent:

49. Application 3: Neural coding: function approximation with tuning curves

51. “Classical view”: multiple spatial maps

52. Application 3: function approximation in sensorimotor area In Parietal cortex: Retinotopic cells gain modulated by eye position And also head position, arm position … Snyder and Pouget, 2000

55. Multisensory integration = multidirectional coordinate transform Experimental validation Model prediction: Pouget, Duhamel and Deneve, 2004 Avillac et al, 2005 Partially shifting tuning curves

56. Unsupervised learning …. First example of many

57. Principal component analysis Orthogonal basis

58. Principal component analysis (unsupervised learning) Orthogonal basis

59. Principal component analysis Orthogonal basis:Uncorrelated components: Note: not the same as independent

60. Principal component analysis and dimensionality reduction K<<N + “Noise”

61. Principal component analysis (unsupervised learning) Orthogonal basis N=2 K=1

62. One solution: eigenvalue decomposition of covariance matrix D D

64. How do we “learn” the parameters? K<<N Standard iterative method First component: other components:

65. PCA: gradient descent « Maximization » « Expectation » Generalized Oja rule

70. Natural images: Weights learnt by PCA

71. Application of PCA: analysis of large neural datasets Machens, Brody and Romo, 2010

72. Application of PCA: analysis of large neural datasets Time Frequency Machens, Brody and Romo, 2010