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Neural networks Eric Postma IKAT Universiteit Maastricht

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Overview Introduction: The biology of neural networks the biological computer brain-inspired models basic notions Interactive neural-network demonstrations Perceptron Multilayer perceptron Kohonen’s self-organising feature map Examples of applications

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A typical AI agent

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Two types of learning Supervised learning Supervised learning –curve fitting, surface fitting,... Unsupervised learning Unsupervised learning –clustering, visualisation...

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An input-output function

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Fitting a surface to four points

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(Artificial) neural networks The digital computer versus the neural computer

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The Von Neumann architecture

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The biological architecture

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Digital versus biological computers 5 distinguishing properties speed speed robustness robustness flexibility flexibility adaptivity adaptivity context-sensitivity context-sensitivity

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Speed: The “hundred time steps” argument The critical resource that is most obvious is time. Neurons whose basic computational speed is a few milliseconds must be made to account for complex behaviors which are carried out in a few hudred milliseconds (Posner, 1978). This means that entire complex behaviors are carried out in less than a hundred time steps. Feldman and Ballard (1982)

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Graceful Degradation damage performance

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Flexibility: the Necker cube

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vision = constraint satisfaction

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Adaptivitiy processing implies learning in biological computers versus processing does not imply learning in digital computers

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Context-sensitivity: patterns emergent properties

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Robustness and context-sensitivity coping with noise

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The neural computer Is it possible to develop a model after the natural example? Is it possible to develop a model after the natural example? Brain-inspired models: Brain-inspired models: –models based on a restricted set of structural en functional properties of the (human) brain

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The Neural Computer (structure)

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Neurons, the building blocks of the brain

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Neural activity in out

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Synapses, the basis of learning and memory

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Learning: Hebb’s rule neuron 1synapseneuron 2

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Connectivity An example: The visual system is a feedforward hierarchy of neural modules Every module is (to a certain extent) responsible for a certain function

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(Artificial) Neural Networks Neurons Neurons –activity –nonlinear input-output function Connections Connections –weight Learning Learning –supervised –unsupervised

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Artificial Neurons input (vectors) input (vectors) summation (excitation) summation (excitation) output (activation) output (activation) a = f(e) e i1i1 i2i2 i3i3

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Input-output function nonlinear function: nonlinear function: e f(e) f(x) = 1 + e -x/a 1 a 0 a

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Artificial Connections (Synapses) w AB w AB –The weight of the connection from neuron A to neuron B AB w AB

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The Perceptron

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Learning in the Perceptron Delta learning rule Delta learning rule –the difference between the desired output t and the actual output o, given input x Global error E Global error E –is a function of the differences between the desired and actual outputs

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Gradient Descent

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Linear decision boundaries

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The history of the Perceptron Rosenblatt (1959) Rosenblatt (1959) Minsky & Papert (1961) Minsky & Papert (1961) Rumelhart & McClelland (1986) Rumelhart & McClelland (1986)

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The multilayer perceptron inputhiddenoutput

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Training the MLP supervised learning supervised learning –each training pattern: input + desired output –in each epoch: present all patterns –at each presentation: adapt weights –after many epochs convergence to a local minimum

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phoneme recognition with a MLP input: frequencies Output: pronunciation

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Non-linear decision boundaries

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Compression with an MLP the autoencoder

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hidden representation

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Learning in the MLP

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Preventing Overfitting GENERALISATION = performance on test set GENERALISATION = performance on test set Early stopping Early stopping Training, Test, and Validation set Training, Test, and Validation set k-fold cross validation k-fold cross validation –leaving-one-out procedure

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Image Recognition with the MLP

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Hidden Representations

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Other Applications Practical Practical –OCR –financial time series –fraud detection –process control –marketing –speech recognition Theoretical Theoretical –cognitive modeling –biological modeling

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Some mathematics…

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Perceptron

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Derivation of the delta learning rule Target output Actual output h = i

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MLP

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Sigmoid function May also be the tanh function May also be the tanh function –( instead of ) Derivative f’(x) = f(x) [1 – f(x)] Derivative f’(x) = f(x) [1 – f(x)]

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Derivation generalized delta rule

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Error function (LMS)

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Adaptation hidden-output weights

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Adaptation input-hidden weights

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Forward and Backward Propagation

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Decision boundaries of Perceptrons Straight lines (surfaces), linear separable

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Decision boundaries of MLPs Convex areas (open or closed)

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Decision boundaries of MLPs Combinations of convex areas

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Learning and representing similarity

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Alternative conception of neurons Neurons do not take the weighted sum of their inputs (as in the perceptron), but measure the similarity of the weight vector to the input vector Neurons do not take the weighted sum of their inputs (as in the perceptron), but measure the similarity of the weight vector to the input vector The activation of the neuron is a measure of similarity. The more similar the weight is to the input, the higher the activation The activation of the neuron is a measure of similarity. The more similar the weight is to the input, the higher the activation Neurons represent “prototypes” Neurons represent “prototypes”

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Course Coding

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2nd order isomorphism

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Prototypes for preprocessing

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Kohonen’s SOFM (Self Organizing Feature Map) Unsupervised learning Unsupervised learning Competitive learning Competitive learning output input (n-dimensional) winner

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Competitive learning Determine the winner (the neuron of which the weight vector has the smallest distance to the input vector) Determine the winner (the neuron of which the weight vector has the smallest distance to the input vector) Move the weight vector w of the winning neuron towards the input i Move the weight vector w of the winning neuron towards the input i Before learning i w After learning i w

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Kohonen’s idea Impose a topological order onto the competitive neurons (e.g., rectangular map) Impose a topological order onto the competitive neurons (e.g., rectangular map) Let neighbours of the winner share the “prize” (The “postcode lottery” principle.) Let neighbours of the winner share the “prize” (The “postcode lottery” principle.) After learning, neurons with similar weights tend to cluster on the map After learning, neurons with similar weights tend to cluster on the map

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Topological order neighbourhoods Square Square –winner (red) –Nearest neighbours Hexagonal Hexagonal –Winner (red) –Nearest neighbours

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A simple example A topological map of 2 x 3 neurons and two inputs A topological map of 2 x 3 neurons and two inputs 2D input input weights visualisation

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Weights before training

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Input patterns (note the 2D distribution)

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Weights after training

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Another example Input: uniformly randomly distributed points Input: uniformly randomly distributed points Output: Map of 20 2 neurons Output: Map of 20 2 neurons Training Training –Starting with a large learning rate and neighbourhood size, both are gradually decreased to facilitate convergence

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Dimension reduction

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Adaptive resolution

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Application of SOFM Examples (input)SOFM after training (output)

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Visual features (biologically plausible)

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Principal Components Analysis (PCA) Principal Components Analysis (PCA) pca1 pca2 pca1 pca2 Projections of data Relation with statistical methods 1

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Relation with statistical methods 2 Multi-Dimensional Scaling (MDS) Multi-Dimensional Scaling (MDS) Sammon Mapping Sammon Mapping Distances in high- dimensional space

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Image Mining the right feature

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Fractal dimension in art Jackson Pollock (Jack the Dripper)

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Taylor, Micolich, and Jonas (1999). Fractal Analysis of Pollock’s drip paintings. Nature, 399, 422. (3 june). Creation date Fractal dimension } Range for natural images

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Our Van Gogh research Two painters Vincent Van Gogh paints Van Gogh Vincent Van Gogh paints Van Gogh Claude-Emile Schuffenecker paints Van Gogh Claude-Emile Schuffenecker paints Van Gogh

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Sunflowers Is it made by Is it made by –Van Gogh? –Schuffenecker?

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Approach Select appropriate features (skipped here, but very important!) Select appropriate features (skipped here, but very important!) Apply neural networks Apply neural networks

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van Gogh Schuffenecker

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Training Data Van Gogh (5000 textures) Schuffenecker (5000 textures)

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Results Generalisation performance Generalisation performance 96% correct classification on untrained data 96% correct classification on untrained data

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Resultats, cont. Trained art-expert network applied to Yasuda sunflowers Trained art-expert network applied to Yasuda sunflowers 89% of the textures is geclassificeerd as a genuine Van Gogh 89% of the textures is geclassificeerd as a genuine Van Gogh

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A major caveat… Not only the painters are different… Not only the painters are different… …but also the material and maybe many other things… …but also the material and maybe many other things…

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