Activations, attractors, and associators

Jaap MurreUniversiteit van Amsterdam en

Universiteit Utrechtmurre@psy.uva.nl

Toets

• Op welke wijze abstraheert een neuraal network neuron (‘node’) van een biologisch neuron?

• Noem tenminste 5 kenmerken

Overview

• Interactive activation model• Hopfield networks• Constraint satisfaction• Attractors• Traveling salesman problem• Hebb rule and Hopfield networks• Bidirectional associative networks• Linear associative networks

Much of perception is dealing with ambiguity

LAB

Many interpretations are processed in parallel

CAB

The final interpretation must satisfy many constraintsIn the recognition of letters and words:i. Only one word can occur at a given positionii. Only one letter can occur at a given

positioniii. A letter-on-a-position activates a wordiv. A feature-on-a-position activates a letter

i. Only one word can occur at a given position

LAP CAP CAB

L.. C.. .A. ..P ..B

ii. Only one letter can occur at a given position

L.. C.. .A. ..P ..B

LAP CAP CAB

iii. A letter-on-a-position activates a word

L.. C.. .A. ..P ..B

LAP CAP CAB

iv. A feature-on-a-position activates a letter

L.. C.. .A. ..P ..B

LAP CAP CAB

L.. C.. .A. ..P ..B

LAP CAP CAB

Recognition of a letter is a process of constraint satisfaction

L.. C.. .A. ..P ..B

LAP CAP CAB

Recognition of a letter is a process of constraint satisfaction

L.. C.. .A. ..P ..B

LAP CAP CAB

Recognition of a letter is a process of constraint satisfaction

L.. C.. .A. ..P ..B

LAP CAP CAB

Recognition of a letter is a process of constraint satisfaction

L.. C.. .A. ..P ..B

LAP CAP CAB

Recognition of a letter is a process of constraint satisfaction

Hopfield (1982)

• Bipolar activations – -1 or 1

• Symmetric weights (no self weights) – wij= wji

• Asynchronous update rule– Select one neuron randomly and update it

• Simple threshold rule for updating

Energy of a Hopfield network

Energy E = - ½ i,jwjiaiaj

E = - ½ i(wjiai + wijai)aj = - iwjiai aj

Net input to node j is iwjiai = netj

Thus, we can write E = - netj aj

Given a net input, netj, find aj so that - netjaj is minimized

• If netj is positive set aj to 1

• If netj is negative set aj to -1

• If netj is zero, don’t care (leave aj as is)• This activation rule ensures that the energy

never increases• Hence, eventually the energy will reach a

minimum value

Attractor• An attractor is a stationary network state

(configuration of activation values)• This is a state where it is not possible to minimize

the energy any further by just flipping one activation value

• It may be possible to reach a deeper attractor by flipping many nodes at once

• Conclusion: The Hopfield rule does not guarantee that an absolute energy minimum will be reached

Attractor

Local minimum

Global minimum

Example: 8-Queens problem

• Place 8 queens on a chess board such that they are not able to take each other

• This implies the following three constraints:– 1 queen per column– 1 queen per row– 1 queen on any diagonal

• This encoding of the constraints ensures that the attractors of the network correspond to valid solutions

The constraints are satisfied by inhibitory connections

Column

Row

Diagonals

Diagonals

Problem: how to ensure that exactly 8 nodes are 1?• A term may be added to control for this in

the activation rule• Binary nodes may be used with a bias• It is also possible to use continuous valid

nodes with Hopfield networks (e.g, between 0 and 1)

Traveling Salesman Problem

The energy minimization question can also be turned around

• Given ai and aj, how should we set the weight wji = wji so that the energy is minimized?

E = - ½ wjiaiaj, so that – when aiaj = 1, wji must be positive

– when aiaj = -1, wji must be negative

• For example, wji= aiaj, where is a learning constant

Hebb and Hopfield

• When used with Hopfield type activation rules, the Hebb learning rule places patterns at attractors

• If a network has n nodes, 0.15n random patterns can be reliably stored by such a system

• For complete retrieval it is typically necessary to present the network with over 90% of the original pattern

Bidirectional Associative Memories (BAM, Kosko 1988)• Uses binary nodes (0 or 1)• Symmetric weights• Input and output layer• Layers are updated in order,

using threshold activation rule

• Nodes within a layer are updated synchronously

BAM

• BAM is in fact a Hopfield network with two layers of nodes

• Within a layer, weights are 0• These neurons are not dependent on each other (no

mutual inputs)• If updated synchronously, there is therefore no

danger of increasing the network energy• BAM is similar to the core of Grossberg’s Adaptive

Resonance Theory (Lecture 4)

Linear Associative Networks

• Invented by Kohonen (1972), Nakano (1972), and by Anderson (1972)

• Two layers• Linear activation rule

– Activation is equal to net input• Can store patterns• Their behavior is mathematically tractable

using matrix algebra

Associating an input vector p with an output vector q

Storage: W = qpT

with = (pTp)-1

Recall: Wp = qpTp = pTpq = q

Inner product pTp gives a scalar

3 0 1 4 0 1301401

p

pT

9 0 1 16 0 1

9 0 116 0 1

27 = (pTp)-1 = 1/27

Outer product qpT gives a matrix

3 0 1 4 0 1120241

3 0 1 4 0 1 6 0 2 8 0 2 0 0 0 0 0 0 6 0 2 8 0 212 0 4 16 0 4 3 0 1 4 0 1

q output vector

pT input vector

W/ weight matrix divided by constant

Final weight matrix W = qpT

3 0 1 4 0 10.11 0 0.04 0.15 0 0.04 10.22 0 0.07 0.3 0 0.07 2

0 0 0 0 0 0 00.22 0 0.07 0.3 0 0.07 20.44 0 0.15 0.59 0 0.15 40.11 0 0.04 0.15 0 0.04 1

Recall: Wp = q

1 0.11 0 0.04 0.15 0 0.04 3

2 0.22 0 0.07 0.3 0 0.07 0

0 0 0 0 0 0 0 1

2 0.22 0 0.07 0.3 0 0.07 4

4 0.44 0 0.15 0.59 0 0.15 0

1 0.11 0 0.04 0.15 0 0.04 1

0.113 + 00 + 0.04 1 + 0.154 + 0 0 + 0.041 = 10.223 + 00 + 0.07 1 + 0.304 + 0 0 + 0.071 = 2

Weight matrix Input vectorOutput vector

Storing n patterns

Storage: Wk = kqkpkT, with k = pk

Tpk

W = W1 + W2 + … + Wk + … + Wn

Recall: Wpk = kqkpkTpk + Error = q + Error

Error = W1pk + … + Whpk + … + Wnpk

is 0 only if phTpk for all h k

Conclusion

• LANs work only well, if the input patterns are (nearly) orthogonal

• If an input pattern overlaps with others, then recall will be contaminated with the output patterns of those overlapping patterns

• It is, therefore, important that input patterns are orthogonal (i.e., have little overlap)

LANs have limited representational power• For each three-layer LAN,

there exists an equivalent two layer LAN

• Proof: Suppose that q = Wp and r = Vq, than we have

r = Vq = VWp = Xpwith X = VW

p

q

r

W

V

p

r

X

Summing up

• There is a wide variety of ways to store and retrieve patterns in neural networks based on the Hebb rule– Willshaw network (associator)– BAM– LAN– Hopfield network

• In Hopfield networks, stored patterns can be viewed as attractors

Summing up

• Finding an attractor is a process of constraint satisfaction. It can can be used as:– A recognition model– A memory retrieval model– A way of solving the traveling salesman

problem and other difficult problems