Machine Learning

Winner Takes All Networks (WTA)

Prof. Dr. Volker Sperschneider

AG Maschinelles Lernen und Natürlichsprachliche Systeme

Institut für Informatik Technische Fakultät

Albert-Ludwigs-Universität Freiburg

sperschneider@informatik.uni-freiburg.de

Motivation •  Unsupervised learning, clustering, prototyping •  Training set of real-valued n-dimensional vectors

(= „blue“ dots in next diagram)

•  Determine k vectors in n-dimensional space (“red“ dots in next diagram) that best represent all training vectors

nlxx ℜ∈,,1

nkww ℜ∈,,1

Motivation

3w

px

pxwdist −= 3

2w1w

Complexity of Clustering

•  Given point set

•  Given allowed number of clusters k.

•  Problem 1: n = 2, l and k variable

Finding optimal clustering is NP-hard.

nlxxx ℜ∈,,, 21

Complexity of Clustering

•  Given point set

•  Given allowed number of clusters k.

•  Problem 2: k = 2, l and n variable

Finding optimal clustering is NP-hard.

nlxxx ℜ∈,,, 21

Complexity of Clustering

•  Given point set

•  Given allowed number of clusters k.

•  Problem 3: k = 2, n fixed, l variable Finding optimal clustering is polynomially solvable with polynomial of degree

nlxxx ℜ∈,,, 21

)log( 1 llO nk+

Motivation •  Assume that norm of training vectors x has

been normalized to 1 on average. •  Therefore we may assume that norm of

prototype vectors w can be fixed to 1, too.

•  Shows that distance minimization is equivalent to inner product maximization.

22

22

22

)()(),(

xxwwxxxwww

xwxwxwxwdistTTTT

T

+−=+−

=−−=−=

Motivation •  Inner products are computed as net input of

neurons. Linear neuron also output this as activation value.

•  Inner product of vectors measure their similarity.

•  Using k linear output neurons by taking the one that outputs maximum activation (= inner product) selects the one whose weight vector maximally resembles the input vector: THE WINNER

Motivation Thus the typical WTA-architecture looks as follows – here 6 input neurons and 4 output neurons used:

1wxT1w2w3w4w

x2wxT3wxT4wxT

Sometimes output neurons are arranged in a specific manner with neighborhood relations between them – an example with 2 input neurons and 15 = 5 x 3 output neurons arranged in a 2-dimensional grid:

Or as a ring structure

Another arrangement is the hexagonal grid:

Neighborhoods are useful when prototype vectors carry symbolic labels and identical labels occur in local neighborhoods:

•  Neighborhoods are useful when neighbours are prototypes for similar tactile (e.g.) sensory inputs - inner topology preserving map of body surface:

•  Map of face with eyes, nose and mouth:

•  Linearly arranged output neurons may become prototypes for city locations and define a short tour – remember TSP

•  Note that topology preservation is only in one

direction: Neurons far away in the line may nevertheless represent close vectors.

•  2-dimenional grid may be trained on triples (x,y,z) with z = f(x,y)) to span the graph of a 2-dimensio- nal real function f in a topology preserving manner – later used for function approximation, and inverse kinematics

•  After training the net, fresh function values are

recovered as follows: Given (x,y), determine the winner neuron of the net with all third weights omitted; then read out function value f(x,y) as third weight of the winner neuron.

x y

z

x y

z

x y

z

Pole balancing

Keep a pole in balance by applying an acceleration (force) depending on angle θ measuring deviation from vertival position, and derivative v of θ by time that measures of how fast angle θ changes.

force force

θ θ

Pole balancing The correct control function is, using appropriate constants a and b: A 2-dimensional grid as Kohonen-net is trained to map the graph of function f, that is, to map to the surface of points in 3-dimensional space

dtd

dtd baf θθ θθ += sin),(

)),(,,( vfv θθ

Inverse kinematics

α

β

x

y (x,y)

Every pair α, β of angles uniquely determines a

pair of coordinates x, y of a point (formulas?).

Inverse kinematics

•  A 2-dimensional Kohonen grid is trained to map the „surface“ consisting of all points

•  Let destination point be given:

)),(),,(,,( βαβαβα yx

),( yx

Inverse kinematics

•  Neuron is determined with weight vector

such that is closest •  Delivered angles are:

),,,( yx wwww βα

),( yx ww ),( yx

),( βα ww

Inverse kinematics

•  Smooth navigation from source to destination: •  Obstacles are avoided provided the Kohonen

grid was trained to map admissible 4-tuples.

•  Technical details and further illustrations taken from now on from Martin Riedmillers pretty presentation

wtan.printer