Un Supervised Learning & Self Organizing Maps

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

( grandmother cells ) units

UnSupervised Competitive Learning

• In the hebbian like models, all the neurons can fire together • In

Competitive Learning

models, only one unit (or one per group) can fire at a time • Output units compete with one another 

Winner Takes All

units (“grandmother cells”)

US Competitive, Cntd

• Such networks cluster the data points • The number of clusters is not predefined but is limited to the number of output units • Applications include VQ, medical diagnosis, document classification and more

Simple Competitive Learning

N inputs units P output neurons P x N weights

h i



i

 1 , 2 ...

P j N

  1

W ij X j

x 1 x 2

Y i

 1 or 0 x N W P1 W 11 W 12 W 22 W PN Y 1 Y 2 Y P

Simple Model, Cntd

• All weights are positive and normalized • Inputs and outputs are binary

i h

*

i

  

W ij j

arg

X j

 max(

h i



W i

)

X

Network Activation

• The unit with the highest field

h

i

fires •

i*

is the winner unit 

W i

* input vector • The winning unit’s weight vector is updated to be even closer to the current input vector • Possible variation: adding lateral inhibition

Learning

Starting with small random weights, at each step: 1. a new input vector is presented to the network 2. all fields are calculated to find a winner  3.

W i

* is updated to be closer to the input

Learning Rule

•

Standard Competitive Learning



W i

*

j

  (

X j

 

W i

*

j

) Can be formulated as hebbian : 

W ij

 

O i

(

X j

 

W ij

)

Result

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

clustering

Competitive Learning, Cntd

• It is important to break the symmetry in the initial random weights • Final configuration depends on initialization – A winning unit has more chances of winning the next time a similar input is seen – Some outputs may never fire – This can be compensated by updating the non winning units with a smaller update

Model: Horizontal & Vertical lines

Rumelhart & 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

Rumelhart & Zipser, Cntd H V

Geometrical Interpretation

• So far the ordering of the output units themselves was not necessarily informative • The location of the winning unit can give us information regarding similarities in the data • We are looking for an input output mapping that

conserves the topologic properties



feature mapping

of the inputs • Given any two spaces, it is not guaranteed that such a mapping exits!

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

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

Self Organizing (Kohonen) Maps

• Competitive networks (WTA neurons) • Output neurons are placed on a lattice, usually 2 dimensional • Neurons become

selectively tuned

patterns (stimuli) to various input • The location of the tuned (winning) neurons become ordered in such a way that creates a

meaningful coordinate system

features  for different input a

topographic map

of input patterns is formed

SOMs, Cntd

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

Self Organization

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



W ij

  F (

i

,

i

*)(

X



j



W ij

)

Neighborhood Function

• F is maximal for

i

* and drops to zero far from

i

, for example: F

(i,i*)

 exp( 

r

i

  2 

r



i

* 2 2 ) • The update “pulls” the winning unit (weight vector) to be closer to the input, and also drags the close neighbors of this unit 

• 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 • Both η and σ can be changed during the learning

Feature Mapping

Weight Vectors Weight Vectors 6 6 3 4 4 2 2 2 1 0 0 0 -2 -2 -1 -4 -4 -2 -6 -6 -3 -4 -4 -6 -4 -2 -2 -4 -2 0 0 0 2 2 4 4 6 6 6 8 8 8 10 10 10 12 12 12 14

Topologic Maps in the Brain

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

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

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

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

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

Summary

• Unsupervised learning is very common • US learning requires redundancy in the stimuli • Self organization is a basic property of the brain’s 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