Activations, attractors, and associators Jaap Murre Universiteit van Amsterdam

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 constraints In the recognition of letters and words: i. Only one word can occur at a given position ii. Only one letter can occur at a given position iii. A letter-on-a-position activates a word iv. A feature-on-a-position activates a letter

Letter detection Press the left button when the letter L is present Press as fast as you can

C##

L##

LAP

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

i. Only one word can occur at a given position LAPCAPCAB L..C...A...P..B

ii. Only one letter can occur at a given position L..C...A...P..B LAPCAPCAB

iii. A letter-on-a-position activates a word L..C...A...P..B LAPCAPCAB

iv. A feature-on-a-position activates a letter L..C...A...P..B LAPCAPCAB

L..C...A...P..B LAPCAPCAB Recognition of a letter is a process of constraint satisfaction

L..C...A...P..B LAPCAPCAB Recognition of a letter is a process of constraint satisfaction

L..C...A...P..B LAPCAPCAB Recognition of a letter is a process of constraint satisfaction

L..C...A...P..B LAPCAPCAB Recognition of a letter is a process of constraint satisfaction

L..C...A...P..B LAPCAPCAB Recognition of a letter is a process of constraint satisfaction

Hopfield (1982) Bipolar activations –-1 or 1 Symmetric weights (no self weights) –w ij = w ji Asynchronous update rule –Select one neuron randomly and update it Simple threshold rule for updating

Energy of a Hopfield network Energy E = - ½  i,j w ji a i a j  E = - ½  i (w ji a i + w ij a i )  a j = -  i w ji a i  a j Net input to node j is  i w ji a i = net j Thus, we can write  E = - net j  a j

Given a net input, net j, find a j so that -net j a j is minimized If net j is positive set a j to 1 If net j is negative set a j to -1 If net j is zero, don’t care (leave a j 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

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 a i and a j, how should we set the weight w ji = w ji so that the energy is minimized?  E = - ½   w ji a i a j, so that –when a i a j = 1,  w ji must be positive –when a i a j = -1,  w ji must be negative For example,  w ji =  a i a j, 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 =  qp T with  = (p T p) -1 Recall: Wp =  qp T p =  p T pq = q

Inner product p T p gives a scalar p pTpT  = (p T p) -1 = 1/27

Outer product qp T gives a matrix q output vector p T input vector W  weight matrix multiplied by learning constant

Final weight matrix W =  qp T

Recall: Wp = q 0.11      1 =      1 = 2 Weight matrix Input vector Output vector

Storing n patterns Storage: W k =  k q k p k T, with  k = p k T p k W = W 1 + W 2 + … + W k + … + W n Recall: Wp k =  k q k p k T p k + Error = q + Error Error = W 1 p k + … + W h p k + … + W n p k is 0 only if p h T p k = 0 for all h  k

Characteristics of LANs 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 = Xp with 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