Activations, attractors, and associators

Activations, attractors, and associators Jaap Murre Universiteit van Amsterdam jaap@murre.com

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 i. Only one word can occur at a given position LAP CAP CAB

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

iii. A letter-on-a-position activates a word LAP CAP CAB L.. C.. .A. ..P ..B

LAP CAP CAB L.. C.. .A. ..P ..B iv. A feature-on-a-position activates a letter

Recognition of a letter is a process of constraint satisfaction LAP CAP CAB L.. C.. .A. ..P ..B

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 Diagonals 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 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 pT 3 0 1 4 0 1 3 0 1 4 0 1 9 0 1 16 0 1 9 0 1 16 0 1 p  = (pTp)-1 = 1/27 27

Outer product qpT gives a matrix pT input vector 3 0 1 4 0 1 3 0 1 4 0 1 6 0 2 8 0 2 0 0 0 0 0 0 6 0 2 8 0 2 12 0 4 16 0 4 3 0 1 4 0 1 1 2 0 2 4 1 q output vector W weight matrix multiplied by learning constant

Final weight matrix W = qpT

Recall: Wp = q Input vector Output vector Weight matrix 0.113 + 00 + 0.04 1 + 0.154 + 0 0 + 0.041 = 1 0.223 + 00 + 0.07 1 + 0.304 + 0 0 + 0.071 = 2

Storing n patterns Storage: Wk = kqkpkT, with k =pkTpk W = W1 + W2 + … + Wk + … + Wn Recall: Wpk = kqkpkTpk + Error = q + Error Error = W1pk + … + Whpk + … + Wnpk is 0 only if phTpk= 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)

r LANs have limited representational power V q • 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 W p r X p

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

Activations, attractors, and associators