Transcript Document

Un Supervised Learning
&
Self Organizing Maps
Learning From Examples
1
1
3
4
5
9
16
6
25
2
36
4
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
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
Hopfield Nets
 Associative Nets (Hopfield like) store
predefined memories.
 During learning, the net goes over all
patterns to be stored (Hebb Rule):
Wij 
1
N
μ
i
X X
μ
μ
j
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”)
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
Self Organization
 Network Organization is fundamental to
the brain
 Functional structure
 Layered structure
 Both parallel processing and serial processing
require organization of the brain
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 neuron’s immediate
environment
until a final configuration develops
Questions
• How can a useful configuration develop
from self organization?
• Can random activity produce coherent
structure?
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
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
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
Principles of Self Organization
1.
2.
3.
4.
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
3
2
1
0
-1
-2
-3
-4
-6
-4
-2
0
2
4
6
8
10
12
14
Redundancy
• Unsupervised learning depends on
redundancy in the data
• Learning is based on finding patterns and
extracting features from the data
Un Supervised Hebbian Learning
• A linear unit: V  W j X j
j
•The learning rule is Hebbian like:
W j  V ( X j  VW j )
The change in weight depends on the product of the neuron’s
output and input, with a term that makes the weights decrease
US Hebbian Learning, Cntd
• Such a net converges into a weight vector
2
that maximizes the average on V
• 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
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
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)
Simple Competitive Learning
N inputs units
P output neurons
P x N weights
x1
W11
W12
x2
W22
WP1
N
hi   Wij X j
Y1
Y2
j 1
i  1, 2... P
Yi  1or 0
xN
WPN
YP
Network Activation
• The unit with the highest field hi fires
• i* is the winner unit

• Geometrically W i* is closest to the current
input vector
• The winning unit’s weight vector is updated
to be even closer to the current input vector
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
Result
• Each output unit moves to the center of
mass of a cluster of input vectors 
clustering
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
Self Organizing (Kohonen) Maps
• Competitive networks (WTA neurons)
• Output neurons are placed on a lattice, usually 2dimensional
• 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
SOMs, Cntd
• Spatial locations of the neurons in the map
are indicative of statistical features that are
present in the inputs (stimuli) 
Self Organization
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
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
Feature Mapping
Weight Vectors
Weight Vectors
6
6
4
4
2
W(i,2)
W(i,2)
2
0
0
-2
-2
3
2
1
0
-1
-4
-4
-2
-6
-6
-3
-4
-4
-4
-6
-2
0
-2
-4
2
0
-2
2
0
4
4
W(i,1)
W(i,1)
2
4
6
6
8
8
6
10
10
8
10
12
12
12
14
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)
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