feedback networks and associative memories

Feedback Networksand Associative Memories

虞台文大同大學資工所智慧型多媒體研究室

Content

IntroductionDiscrete Hopfield NNsContinuous Hopfield NNsAssociative Memories

– Hopfield Memory– Bidirection Memory


Introduction

大同大學資工所智慧型多媒體研究室

Feedforward/Feedback NNs

Feedforward NNs– The connections between units do not form cycles.– Usually produce a response to an input quickly.– Most feedforward NNs can be trained using a wide variet

y of efficient algorithms. Feedback or recurrent NNs

– There are cycles in the connections. – In some feedback NNs, each time an input is presented, t

he NN must iterate for a potentially long time before it produces a response.

– Usually more difficult to train than feedforward NNs.

Supervised-Learning NNs

Feedforward NNs– Perceptron – Adaline, Madaline – Backpropagation (BP) – Artmap – Learning Vector Quantization (LVQ) – Probabilistic Neural Network (PNN) – General Regression Neural Network (GRNN)

Feedback or recurrent NNs– Brain-State-in-a-Box (BSB) – Fuzzy Congitive Map (FCM) – Boltzmann Machine (BM) – Backpropagation through time (BPTT)

Unsupervised-Learning NNs

Feedforward NNs– Learning Matrix (LM) – Sparse Distributed Associative Memory (SDM) – Fuzzy Associative Memory (FAM) – Counterprogation (CPN)

Feedback or Recurrent NNs– Binary Adaptive Resonance Theory (ART1) – Analog Adaptive Resonance Theory (ART2, ART2a) – Discrete Hopfield (DH) – Continuous Hopfield (CH) – Discrete Bidirectional Associative Memory (BAM) – Kohonen Self-organizing Map/Topology-preserving map

(SOM/TPM)

The Hopfield NNs

In 1982, Hopfield, a Caltech physicist, mathematically tied together many of the ideas from previous research.

A fully connected, symmetrically weighted network where each node functions both as input and output node.

Used for– Associated memories– Combinatorial optimization

Associative Memories

An associative memory is a content-addressable structure that maps a set of input patterns to a set of output patterns.

Two types of associative memory: autoassociative and heteroassociative.

Auto-association– retrieves a previously stored pattern that most closely re

sembles the current pattern. Hetero-association

– the retrieved pattern is, in general, different from the input pattern not only in content but possibly also in type and format.


Auto-association

AA

Hetero-association

Niagara Waterfall

memorymemory

memorymemory

Optimization Problems

Associate costs with energy functions in Hopfield Networks– Need to be in quadratic form

Hopfield Network finds local, satisfactory soluions, doesn’t choose solutions from a set.

Local optimums, not global.


Discrete Hopfield NNs


The Discrete Hopfield NNs

11 22 33 nn. . .

v1 v2 v3 vn

I1 I2 I3 In

w21 w31 wn1

w12 w32 wn2

w13 w23 wn3

w1n w2n w3n

The Discrete Hopfield NNs

11 22 33 nn. . .

v1 v2 v3 vn

I1 I2 I3 In

w21 w31 wn1

w12 w32 wn2

w13 w23 wn3

w1n w2n w3n

wij = wji

wii = 0

wij = wji

wii = 0

State Update Rule

11 22 33 nn. . .

v1 v2 v3 vn

I1 I2 I3 In

w21 w31 wn1

w12 w32 wn2

w13 w23 wn3

w1n w2n w3n

11 22 33 nn. . .11 22 33 nn. . .

v1 v2 v3 vnv1 v2 v3 vn

I1 I2 I3 InI1 I2 I3 InI1 I2 I3 In

w21 w31 wn1w21 w31 wn1

w12 w32 wn2w12 w32 wn2

w13 w23 wn3w13 w23 wn3

w1n w2n w3nw1n w2n w3n

Asynchronous mode

Update rule

1

() )( 1 i

n

i ij jjj i

v t IH t w

1 0( 1) sg

( 1)(

1 0(n 1)

1)i

iii

H tH t

H tv t

Stable?Stable?

Energy Function

11 22 33 nn. . .

v1 v2 v3 vn

I1 I2 I3 In

w21 w31 wn1

w12 w32 wn2

w13 w23 wn3

w1n w2n w3n

11 22 33 nn. . .11 22 33 nn. . .



w21 w31 wn1w21 w31 wn1

w12 w32 wn2w12 w32 wn2

w13 w23 wn3w13 w23 wn3


1

() )( 1 i

n

i ij jjj i

v t IH t w

( 1)

( 1

1 0( 1)

1 0)ii

i

H t

tv

Ht

12

1 1 1

n n n

i j ii j i

i ij IE w v v v

Fact:E is lower bounded (upper bounded).

If E is monotonically decreasing (increasing), the system is stable.

The Proof

11 22 33 nn. . .

v1 v2 v3 vn

I1 I2 I3 In

w21 w31 wn1

w12 w32 wn2

w13 w23 wn3

w1n w2n w3n

11 22 33 nn. . .11 22 33 nn. . .



w21 w31 wn1w21 w31 wn1

w12 w32 wn2w12 w32 wn2

w13 w23 wn3w13 w23 wn3


1

() )( 1 i

n

i ij jjj i

v t IH t w

( 1)

( 1

1 0( 1)

1 0)ii

i

H t

tv

Ht

12

1 1 1

n n n

i j ii j i

i ij IE w v v v

Suppose that at time t + 1, the kth neuron is selected for update.

12

1 1 1 1

( ) ( ) ( ) ( ) ( ) ( ) ( )n n n n

ij i j i i ik k k k

i

ii j i i

k j k i k

E t w v t v t I v t w v t v t I v t

12

1 1 1 1

( 1) ( 1) ( 1) ( 1) ( 1) ( 1) ( 1)n n n n

ij i j i i i ii j i i

k k k k

i k j k i k


The Proof

11 22 33 nn. . .

v1 v2 v3 vn

I1 I2 I3 In

w21 w31 wn1

w12 w32 wn2

w13 w23 wn3

w1n w2n w3n

11 22 33 nn. . .11 22 33 nn. . .



w21 w31 wn1w21 w31 wn1

w12 w32 wn2w12 w32 wn2

w13 w23 wn3w13 w23 wn3


1

() )( 1 i

n

i ij jjj i

v t IH t w

( 1)

( 1

1 0( 1)

1 0)ii

i

H t

tv

Ht

12

1 1 1

n n n

i j ii j i

i ij IE w v v v

Suppose that at time t + 1, the kth neuron is selected for update.

12

1 1 1 1

( ) ( ) ( ) ( ) ( ) ( ) ( )n n n n

ij i j i i ik k k k

i

ii j i i

k j k i k


12

1 1 1 1

( 1) ( 1) ( 1) ( 1) ( 1) ( 1) ( 1)n n n n


k k k k

i k j k i k


Their values are not changed at

time t + 1.


time t + 1.


time t + 1.


time t + 1.

The Proof

1

() )( 1 i

n

i ij jjj i

v t IH t w

( 1)

( 1

1 0( 1)

1 0)ii

i

H t

tv

Ht

12

1 1 1 1

( ) ( ) ( ) ( ) ( ) ( ) ( )n n n n

ij i j i i ik k k k

i

ii j i i

k j k i k


12

1 1 1 1

( 1) ( 1) ( 1) ( 1) ( 1) ( 1) ( 1)n n n n


k k k k

i k j k i k


1 1

( 1) ( ) ( 1) ( 1) (( ) ( )) ( )n n

k k k ki

k ii

ki i k i kIE E t E t v t v t vw v t w v tt v tI

1

[ ( 1) (( )]) k

n

k i k ki

iw v t t tI v v

( 1)[ ( 1) ( )]k k kH t v t v t

The Proof

1

() )( 1 i

n

i ij jjj i

v t IH t w

( 1)

( 1

1 0( 1)

1 0)ii

i

H t

tv

Ht

1 1

( 1) ( ) ( 1) ( 1) (( ) ( )) ( )n n

k k k ki

k ii

ki i k i kIE E t E t v t v t vw v t w v tt v tI

1

[ ( 1) (( )]) k

n

k i k ki

iw v t t tI v v

( 1)[ ( 1) ( )]k k kH t v t v t

1 0 1 0

1 < 0 1 < 0

1 0 1 < 0

1 < 0 1 0

vk(t) vk(t+1)Hk(t+1) E

E StableStable


Continuous Hopfield NNs


The Neuron of Continuous Hopfield NNs

ui

vi =a(ui)

1

1

vi

ui =a1(vi)

11

...

giCi

wi1

wi2

win

Iiv1

v2

vn

ui

vivi

( )ia u

The Dynamics

...

giCi

wi1

wi2

win

Iiv1

v2

vn

ui

vivi

( )ia u

1 1( )i iuv w

2 2( )i iuv w

( )i inn uv w

idi t

udC iig u

1

( )n

i iji

i ijj i

j i i

dC

uu Iug

dtvw

1 1

n ni

i ij ij i i ij jj i j i

j

dC w w g I

u

dv u

t

G i

1

n

i ij i ijj i

iij

dC w G I

dtuv

u

The Continuous Hopfield NNs

g1C1

I1

u1g2C2

I2

u2g3C3

I3

u3gnCn

In

un

v1 v2 v3 vn

v1 v2 v3 vn

w21 wn1

w31

w12

w13

w1n

w23

w2n

w32

w3n

wn2

wn3

. . .. . .

1

n

i ij i ijj i

iij

dC w G I

dtuv

u

The Continuous Hopfield NNs

g1C1

I1

u1g2C2

I2

u2g3C3

I3

u3gnCn

In

un

v1 v2 v3 vn

v1 v2 v3 vn

w21 wn1

w31

w12

w13

w1n

w23

w2n

w32

w3n

wn2

wn3

. . .. . .

1

n

i ij i ijj i

iij

dC w G I

dtuv

u

nn

n

ijj

njnjn

n

n

ijj

jj

n

ijj

jj

n

ijj

jj

IuGvwdt

duC

IuGvwdt

duC

IuGvwdt

duC

IuGvwdt

duC

1

331

333

3

221

222

2

111

111

1

nn

n

ijj

njnjn

n

n

ijj

jj

n

ijj

jj

n

ijj

jj

IuGvwdt

duC

IuGvwdt

duC

IuGvwdt

duC

IuGvwdt

duC

1

331

333

3

221

222

2

111

111

1

Stable?Stable?

Equilibrium Points

Consider the autonomous system:

Equilibrium Points Satisfy

1 1

2 2

( )

( )

( )n n

y f

y f

y f

y

y

y

( ) ( )t y f y

( )*0 f y

The system is asymptotically stable if the following holds:

Lyapunov Theorem

( ) ( )t y f y

There exists a positive-definite function E(y) s.t.

( )0

dE

dt

y

Call E(y) as energy function.

Lyapunov Energy Function

g1C1

I1

u1g2C2

I2

u2g3C3

I3

u3gnCn

In

un

v1 v2 v3 vn

v1 v2 v3 vn

w21 wn1

w31

w12

w13

w1n

w23

w2n

w32

w3n

wn2

wn3

. . .. . .

1

01 1 1 1

1 1( )

2

in n n n v

i j iij i ii j i i

j n

E v v v a vw dGI v

1

01 1 1 1

1 1( )

2

in n n n v

i j iij i ii j i i

j n

E v v v a vw dGI v

Lyapunov Energy Function

g1C1

I1

u1g2C2

I2

u2g3C3

I3

u3gnCn

In

un

v1 v2 v3 vn

v1 v2 v3 vn

w21 wn1

w31

w12

w13

w1n

w23

w2n

w32

w3n

wn2

wn3

. . .. . .

uva )(1 uva )(1

v

u=a1(v)

11

v1 1

dxxav

0

1 )(

1

01 1 1 1

1 1( )

2

in n n n v

i j i i ii j i i

j

i

n

ijE v Gv v a v dvIw

1

01 1 1 1

1 1( )

2

in n n n v

i j i i ii j i i

j

i

n

ijE v Gv v a v dvIw

ui

vi =a(ui)

1

1

1( )0

da v

dv

1( )

0da v

dv

Stability of Continuous Hopfield NNs

1

ni

i j i ijj i

ij iGdu

C v udt

w I

1

01 1 1 1

1 1( )

2

in n n n v

i j i i ii j i i

j

i

n

ijE v Gv v a v dvIw

1( )i i iu a v 1( )i i iu a v 11 ( )i i iu a v

11 ( )i i iu a v

1( )1i i i i

i

du da v dv

dt dv dt

1( )1i i i i

i

du da v dv

dt dv dt

1

ni

i i

dvdE dE

dt dv dt

1 1

n ni

j ii j

i i

i

j

j

i IGwdv

v udt

1

ni i

ii

du dvC

dt dt

21

1

( )1 ni i i

ii i

da v dvC

dv dt

Dynamics


1

ni

i j i ijj i

ij iGdu

C v udt

w I

1

01 1 1 1

1 1( )

2

in n n n v

i j i i ii j i i

j

i

n

ijE v Gv v a v dvIw

21

1

( )1 ni i i

ii i

da v dvdEC

dt dv dt

Dynamics

> 0

v

u=a1(v)

11

1( )0

da v

dv

0dE

dt0

dE

dt


g1C1

I1

u1g2C2

I2

u2g3C3

I3

u3gnCn

In

un

v1 v2 v3 vn

v1 v2 v3 vn

w21 wn1

w31

w12

w13

w1n

w23

w2n

w32

w3n

wn2

wn3

. . .. . .

nn

n

ijj

njnjn

n

n

ijj

jj

n

ijj

jj

n

ijj

jj

IuGvwdt

duC

IuGvwdt

duC

IuGvwdt

duC

IuGvwdt

duC

1

331

333

3

221

222

2

111

111

1

nn

n

ijj

njnjn

n

n

ijj

jj

n

ijj

jj

n

ijj

jj

IuGvwdt

duC

IuGvwdt

duC

IuGvwdt

duC

IuGvwdt

duC

1

331

333

3

221

222

2

111

111

1

StableStable

Basins of Attraction

Local/Global Minima

Energy Landscape


Also named content-addressable memory.

Autoassociative Memory– Hopfield Memory

Heteroassociative Memory– Bidirection Associative Memory (BAM)


Stored Patterns

1

2

1

2

( , )

( , )

( , )p p

x

x

x

y

y

y

1

2

1

2

( , )

( , )

( , )p p

x

x

x

y

y

y

ni Rxmi Ry

ii x yii x y

Autoassociative

Heteroassociative


Associative Memories Hopfield Memory Bidirection Memory


Hopfield Memory

1210Neurons

Fully connected

14,400 weights

Example

StoredPatterns

MemoryAssociation

Example

StoredPatterns

MemoryAssociation

How to Store Patterns?How to Store Patterns?

The Storage Algorithm

Suppose the set of stored pattern is of dimension n.

nnnn

n

n

www

www

www

21

22221

11211

W

nnnn

n

n

www

www

www

21

22221

11211

W =?


1

( )p

k k T

k

p

W x x I

1 2( , , , ) 1, 2, .k k k k Tnx x x k p x

{ 1, 1} 1, 2, .kix i n

1

0

pk ki j

kij

x x i jw

i j

Analysis1

( )p

k k T

k

p

W x x I 1

0

pk ki j

kij

x x i jw

i j

Suppose that x xi.

1

( )p

k k T

k

p

Wx x x I x

( )i i in p n p x x x

Example

1

2

(1, 1, 1,1)

( 1,1, 1,1)

T

T

x

x

1111

1111

1111

1111

)( 11 Txx

1111

1111

1111

1111

)( 22 Txx

2200

2200

0022

0022

)()( 2211 TT xxxx

0 2 0 0

2 0 0 0

0 0 0 2

0 0 2 0

W

1

( )p

k k T

k

p

W x x I 1

0

pk ki j

kij

x x i jw

i j

Example

1

2

(1, 1, 1,1)

( 1,1, 1,1)

T

T

x

x

0 2 0 0

2 0 0 0

0 0 0 2

0 0 2 0

W

11 22 33 442 2

12

1 1 1

( )n n n

i j ii

iij i

jE w x x xI

x

12

1 1

n n

i jiji j

x xw

1 2 3 42( )x x x x

12

T x Wx

Example

1

2

(1, 1, 1,1)

( 1,1, 1,1)

T

T

x

x

11 22 33 442 2

1 2 3 4( ) 2( )E x x x x x

1 1 11

1 11 1

1 1 1 1

Stable

E=4E=4

E=0E=0

E=4E=4

Example

1

2

(1, 1, 1,1)

( 1,1, 1,1)

T

T

x

x

11 22 33 442 2

1 2 3 4( ) 2( )E x x x x x

1 1 11

1 11 1

11 1 1

Stable

E=4E=4

E=0E=0

E=4E=4

Problems of Hopfield Memory

Complement Memories

Spurious stable states

Capacity

Capacity of Hopfield Memory

The number of storable patterns w.r.t. the size of the network.

Study methods:– Experiments– Probability– Information theory– Radius of attraction ()

Capacity of Hopfield Memory

The number of storable patterns w.r.t. the size of the network.

Hopfield (1982) demonstrated that the maximum number of patterns that can be stored in the Hopfield model of n nodes before the error in the retrieved pattern becomes severe is around 0.15n.

The memory capacity of the Hopfield model can be increased as shown by Andrecut (1972).

Radius of attraction (0 1/2)

n

False memories

n

nc

ln4

)21( 2

n

nc

ln4

)21( 2


Associative Memories Hopfield Memory Bidirection Memory


Bidirection Memory

. . .y1 y2 yn

. .

.

x1 x2 xm

X Layer

Y Layer

W=[wij]nm

(( ) )1t a t xWy

Forw

ard

Pass

Forw

ard

Pass

( )tx

( )ty

Back

ward

Pass

Back

ward

Pass

( 1 ( )) Tt ta x yW

Bidirection Memory

. . .y1 y2 yn

. .

.

x1 x2 xm

X Layer

Y Layer

W=[wij]nm

(( ) )1t a t xWy

Forw

ard

Pass

Forw

ard

Pass

( )tx

( )ty

Back

ward

Pass

Back

ward

Pass

( 1 ( )) Tt ta x yW

Stored Patterns

1

2

1

2

( , )

( , )

( , )p p

x

x

x

y

y

y

1

2

1

2

( , )

( , )

( , )p p

x

x

x

y

y

y

?


StoredPatterns

1

2

1

2

( , )

( , )

( , )p p

x

x

x

y

y

y

1

2

1

2

( , )

( , )

( , )p p

x

x

x

y

y

y

2

1 2

1( , , , )

( , , , )k

T

Tn

km

y

x x

y

x

y

y

x

{ 1,1}ix

{ 1,1}iy

1

( )p

k

k

k T

y xW

1ij

pkk

ik

jxw y

Analysis

Suppose xk’ is one of the stored vector.

1

( )p

k

k k T kka a

x xy yxW

1'

( )p

kk

kk T

k

k kma

x xy y

0

k ka m y y

1

( )p

k

k

k T

y xW

Energy Function

1 12 2( , )

T T T

T

E

x y x W y y Wx

y Wx

feedback networks and associative memories

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