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

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

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

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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).

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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…

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Regression (supervised) Target output Parameters

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Linear classifier ? Training examples

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Linear classifier Decision boundary Heavyside function: 0 1

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Linear classifier Decision boundary Heavyside function: 0 1

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Assumptions Multivariate Gaussians Same covariance Two classes equiprobable

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

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How do we compute the output? Orthogonal to decision boundary

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

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

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

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

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

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

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

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

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

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Application 1: Neural population decoding

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w

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How to find ? w w

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

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

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

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

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

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

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Extension: multilayer perceptron Towards a universal computer Initial error:

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

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

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

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

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

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

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Regression (supervised) Target output Parameters

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Regression in general Target output Basis functions

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Gaussian noise assumption

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How to learn the parameters? Gradient descent:

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But: overfitting...

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How to learn the parameters? Gradient descent:

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Application 3: Neural coding: function approximation with tuning curves

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“Classical view”: multiple spatial maps

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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

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

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Unsupervised learning …. First example of many

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Principal component analysis Orthogonal basis

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Principal component analysis (unsupervised learning) Orthogonal basis

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

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

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

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One solution: eigenvalue decomposition of covariance matrix D D

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

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

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Natural images: Weights learnt by PCA

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

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

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