Un Supervised Learning

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Un Supervised Learning

• Self Organizing Maps

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Learning From Examples
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Supervised Learning

• When a set of targets of interest is provided by
  an external teacher
• we say that the learning is Supervised
• The targets usually are in the form of an input
  output mapping
• that the net should learn

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Feed Forward Nets

• Feed Forward Nets learn under supervision
• classification - all patterns in the training set
  are coupled with the correct classification
• classifying written digits into 10 categories
  (the US post zip code project)
• function approximation the values to be learnt
  for the training points is known
• time series prediction such as weather forecast
  and stock values

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Hopfield Nets

• Associative Nets (Hopfield like) store predefined
  memories.
• During learning, the net goes over all patterns
  to be stored (Hebb Rule)

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Hopfield, Cntd

• When presented with an input pattern that is
  similar to one of the memories, the network
  restores the right memory, previously stored in
  its weights (synapses)

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How do we learn?

• Many times there is no teacher to tell us how
  to do things
• A baby that learns how to walk
• Grouping of events into a meaningful scene
  (making sense of the world)
• Development of ocular dominance and orientation
  selectivity in our visual system

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Self Organization

• Network Organization is fundamental to the brain
• Functional structure
• Layered structure
• Both parallel processing and serial processing
  require organization of the brain

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Self Organizing Networks

• Discover significant patterns or features in the
  input data
• Discovery is done without a teacher
• Synaptic weights are changed according to
• local rules
• The changes affect a neurons immediate
  environment
• until a final configuration develops

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Questions

• How can a useful configuration develop from self
  organization?
• Can random activity produce coherent structure?

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Answer biologically

• There are self organized structures in the brain
• Neuronal networks grow and evolve to be
  computationally efficient both in vitro and in
  vivo
• Random activation of the visual system can lead
  to layered and structured organization

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Answer mathematically

• A. Turing, 1952
• Global order can arise from local interactions
• Random local interactions between neighboring
  neurons can coalesce into states of global order,
  and lead to coherent spatio temporal behavior

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Mathematically, Cntd

• Network organization takes place at 2 levels that
  interact with each other
• Activity certain activity patterns are produced
  by a given network in response to input signals
• Connectivity synaptic weights are modified in
  response to neuronal signals in the activity
  patterns
• Self Organization is achieved if there is
  positive feedback between changes in synaptic
  weights and activity patterns

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Principles of Self Organization

• Modifications in synaptic weights tend to self
  amplify
• Limitation of resources lead to competition among
  synapses
• Modifications in synaptic weights tend to
  cooperate
• Order and structure in activation patterns
  represent redundant information that is
  transformed into knowledge by the network

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Redundancy

• Unsupervised learning depends on redundancy in
  the data
• Learning is based on finding patterns and
  extracting features from the data

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Un Supervised Hebbian Learning

• A linear unit

• The learning rule is Hebbian like

The change in weight depends on the product of
the neurons output and input, with a term that
makes the weights decrease
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US Hebbian Learning, Cntd

• Such a net converges into a weight vector that
  maximizes the average on

• This means that the weight vector points at
  the first principal component of the data
• The network learns a feature of the data without
  any prior knowledge
• This is called feature extraction

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Visual Model

• Linsker (1986) proposed a model of self
  organization in the visual system, based on
  unsupervised Hebbian learning
• Input is random dots (does not need to be
  structured)
• Layers as in the visual cortex, with FF
  connections only (no lateral connections)
• Each neuron receives inputs from a well defined
  area in the previous layer (receptive fields)
• The network developed center surround cells in
  the 2nd layer of the model and orientation
  selective cells in a higher layer
• A self organized structure evolved from (local)
  hebbian updates

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Un Supervised Competitive Learning

• In Hebbian networks, all neurons can fire at the
  same time
• Competitive learning means that only a single
  neuron from each group fires at each time step
• Output units compete with one another.
• These are winner takes all units (grandmother
  cells)

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Simple Competitive Learning
N inputs units P output neurons P x N weights

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Network Activation

• The unit with the highest field hi fires
• i is the winner unit
• Geometrically is closest to the current
  input vector
• The winning units weight vector is updated to be
  even closer to the current input vector

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Learning

• Starting with small random weights, at each
  step
• a new input vector is presented to the network
• all fields are calculated to find a winner
• is updated to be closer to the input

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Result

• Each output unit moves to the center of mass of a
  cluster of input vectors ?
• clustering

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Model Horizontal Vertical linesRumelhart
Zipser, 1985

• Problem identify vertical or horizontal signals
• Inputs are 6 x 6 arrays
• Intermediate layer with 8 WTA units
• Output layer with 2 WTA units
• Cannot work with one layer

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Rumelhart Zipser, Cntd
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Self Organizing (Kohonen) Maps

• Competitive networks (WTA neurons)
• Output neurons are placed on a lattice, usually
  2-dimensional
• Neurons become selectively tuned to various input
  patterns (stimuli)
• The location of the tuned (winning) neurons
  become ordered in such a way that creates a
• meaningful coordinate system for different input
  features ?
• a topographic map of input patterns is formed

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SOMs, Cntd

• Spatial locations of the neurons in the map are
  indicative of statistical features that are
  present in the inputs (stimuli) ?
• Self Organization

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Biological Motivation

• In the brain, sensory inputs are represented by
  topologically ordered computational maps
• Tactile inputs
• Visual inputs (center-surround, ocular dominance,
  orientation selectivity)
• Acoustic inputs

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Biological Motivation, Cntd

• Computational maps are a basic building block of
  sensory information processing
• A computational map is an array of neurons
  representing slightly different tuned processors
  (filters) that operate in parallel on sensory
  signals
• These neurons transform input signals into a
  place coded structure

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Kohonen Maps

• Simple case 2-d input and 2-d output layer
• No lateral connections
• Weight update is done for the winning neuron and
  its surrounding neighborhood
• The output layer is a sort of an elastic net that
  wants to come as close as possible to the inputs
• The output maps conserves the topological
  relationships of the inputs

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Feature Mapping
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Kohonen Maps, Cntd

• Examples of topologic conserving mapping between
  input and output spaces
• Retintopoical mapping between the retina and the
  cortex
• Ocular dominance
• Somatosensory mapping (the homunculus)

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Models

• Goodhill (1993) proposed a model for the
  development of retinotopy and ocular dominance,
  based on Kohonen Maps
• Two retinas project to a single layer of cortical
  neurons
• Retinal inputs were modeled by random dots
  patterns
• Added between eyes correlation in the inputs
• The result is an ocular dominance map and a
  retinotopic map as well

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Models, Cntd

• Farah (1998) proposed an explanation for the
  spatial ordering of the homunculus using a simple
  SOM.
• In the womb, the fetus lies with its hands close
  to its face, and its feet close to its genitals
• This should explain the order of the
  somatosensory areas in the homunculus

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Other Models

• Semantic self organizing maps to model language
  acquisition
• Kohonen feature mapping to model layered
  organization in the LGN
• Combination of unsupervised and supervised
  learning to model complex computations in the
  visual cortex

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Examples of Applications

• Kohonen (1984). Speech recognition - a map of
  phonemes in the Finish language
• Optical character recognition - clustering of
  letters of different fonts
• Angeliol etal (1988) travelling salesman
  problem (an optimization problem)
• Kohonen (1990) learning vector quantization
  (pattern classification problem)
• Ritter Kohonen (1989) semantic maps

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Summary

• Unsupervised learning is very common
• US learning requires redundancy in the stimuli
• Self organization is a basic property of the
  brains computational structure
• SOMs are based on
• competition (wta units)
• cooperation
• synaptic adaptation
• SOMs conserve topological relationships between
  the stimuli
• Artificial SOMs have many applications in
  computational neuroscience

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