Neural networks Eric Postma IKAT Universiteit Maastricht

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

A typical AI agent

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

An input-output function

Fitting a surface to four points

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

The Von Neumann architecture

The biological architecture

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

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)

Graceful Degradation damage performance

Flexibility: the Necker cube

vision = constraint satisfaction

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

Context-sensitivity: patterns emergent properties

Robustness and context-sensitivity coping with noise

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

The Neural Computer (structure)

Neurons, the building blocks of the brain

Neural activity in out

Synapses, the basis of learning and memory

Learning: Hebb’s rule neuron 1synapseneuron 2

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

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

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

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

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

The Perceptron

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

Gradient Descent

Linear decision boundaries

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

The multilayer perceptron inputhiddenoutput

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

phoneme recognition with a MLP input: frequencies Output: pronunciation

Non-linear decision boundaries

Compression with an MLP the autoencoder

hidden representation

Learning in the MLP

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

Image Recognition with the MLP

Hidden Representations

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

Some mathematics…

Perceptron

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

MLP

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)]

Derivation generalized delta rule

Error function (LMS)

Adaptation hidden-output weights

Adaptation input-hidden weights

Forward and Backward Propagation

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

Decision boundaries of MLPs Convex areas (open or closed)

Decision boundaries of MLPs Combinations of convex areas

Learning and representing similarity

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”

Course Coding

2nd order isomorphism

Prototypes for preprocessing

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

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

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

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

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

Weights before training

Input patterns (note the 2D distribution)

Weights after training

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

Dimension reduction

Adaptive resolution

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

Visual features (biologically plausible)

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

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

Image Mining the right feature

Fractal dimension in art Jackson Pollock (Jack the Dripper)

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

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

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

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

van Gogh Schuffenecker

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

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

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

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…