Associative Memories PDF

Associative Memories Outline Introduction to Associative Memories Hopfield Networks Stability Analysis Storage Algorithm Retrieval Phase Bidirectional associative memories (BAM) Associative Memories An associative memory is a content-addressable structure that maps a se...

Associative Memories Outline Introduction to Associative Memories Hopfield Networks Stability Analysis Storage Algorithm Retrieval Phase Bidirectional associative memories (BAM) 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 resembles 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. Associative Memories Auto-association A memory A Hetero-association Niagara Waterfall memory The Hopfield Network Neural networks were designed on analogy with the brain. The brain’s memory, however, works by association. For example, we can recognize a familiar face even in an unfamiliar environment within 100- 200 ms. We can also recall a complete sensory experience, including sounds and scenes, when we hear only a few bars of music. The brain routinely associates one thing with another. Hopfield Network Multilayer neural networks trained with the back- propagation algorithm are used for pattern recognition problems. However, to emulate the human memory’s associative characteristics we need a different type of network: a recurrent neural network. A recurrent neural network has feedback loops from its outputs to its inputs. The presence of such loops has a profound impact on the learning capability of the network. The stability of recurrent networks The stability of recurrent networks intrigued several researchers in the 1960s and 1970s. However, none was able to predict which network would be stable, and some researchers were pessimistic about finding a solution at all. The problem was solved only in 1982, when John Hopfield formulated the physical principle of storing information in a dynamically stable network. Hopfield Networks It is a special type of Dynamic Network (recurrent) It is a single layer feedback network which was first introduced by John Hopfield (1982) Hopfield networks can provide ‒ associations or classifications ‒ optimization problem solution ‒ restoration of patterns ‒ Deal with noisy patterns Architecture: Single-layer Hopfield network Architectural graph of a Hopfield network consisting of N = 4 neurons. The Discrete Hopfield NNs wij = wji wii = 0 w1n w2 n w3n w13 w23 wn3 w12 w32 wn2 w21 w31 wn1 1 2 3... n I1 I2 I3 In v1 v2 v3 vn Hopfield (Activation Function) The Hopfield network uses McCulloch and Pitts neurons with the sign activation function as its computing element: y  1, if v  0  Y   1, if v   sign  Y , if v    State Vector The current state of the Hopfield network is determined by the current outputs of all neurons, y1, y2,..., yn. Thus, for a single-layer n-neuron network, the state can be defined by the state vector as:  y1  y  Y   2       yn   Storage Phase: Weights In the Hopfield network, synaptic weights between neurons are usually represented in matrix form as follows: M  T W  Ym Ym M I m1 where M is the number of states to be memorized by the network, Ym is the n-dimensional binary vector, I is n x n identity matrix, and superscript T denotes a matrix transposition. Possible states for the three-neuron Hopfield network y2 (1, 1, 1) (1, 1, 1) (1, 1, 1) (1, 1, 1) y1 0 (1, 1, 1) (1, 1, 1) (1, 1, 1) (1, 1, 1) y3  The stable state-vertex is determined by the weight matrix W, the current input vector X. If the input vector is partially incorrect or incomplete, the initial state will converge into the stable state-vertex after a few iterations. Suppose, for instance, that our network is required to memorise two opposite states, (1, 1, 1) and (-1, -1, -1). Thus, 1  1 Y1  1 Y2   1 1  1 where Y1 and Y2 are the three-dimensional vectors. Storage Phase we can now determine the weight matrix as follows: M W  m m M I Y Y T m1 1  1 1 0 0 0 2 2 W  11 1 1   1 1  1  1  2 0 1 0  2 0 2 1  1 0 0 1 2 2 0 Next, the network is tested by the sequence of input vectors, X1 and X2, which are equal to the output (or target) vectors Y1 and Y2, respectively. Test First, we activate the Hopfield network by applying the input vector X. Then, we calculate the actual output vector Y, and finally, we compare the result with the initial input vector X.  0 2 2 1  1       Y1  sign 2 0 2 1   1 2 2 0 1  1   0 2 2  1   1        Y2  sign 2 0 2  1    1 2 2 0  1   1  Fundamental memories The remaining six states are all unstable. However, stable states (also called fundamental memories) are capable of attracting states that are close to them. The fundamental memory (1, 1, 1) attracts unstable states (-1, 1, 1), (1, -1, 1) and (1, 1, -1). Each of these unstable states represents a single error, compared to the fundamental memory (1, 1, 1). The fundamental memory (-1, -1, -1) attracts unstable states (-1, -1, 1), (-1, 1, -1) and (1, -1, -1). Thus, the Hopfield network can act as an error correction network. Example2 To illustrate the emergent behavior of the Hopfield model, consider the network which consists of three neurons. The weight matrix of the network is: With three neurons in the network there are 23=8 possible states to consider. Of these eight states, only the two states (1,-1,1) and (-1,1,-1) are stable; the remaining six states are all unstable.  For the state vector (1,-1,1) we have Stability Analysis To evaluate the stability property of the dynamical system of interest, let us study a so-called energy function. This is a function usually defined in n-dimensional output space v 1 t E   v Wv 2 In stability analysis, by defining the structure of W ( symmetric and zero in diagonal terms) and updating method of output (asynchronously) the objective is to show that changing the computational energy in a time duration. If E is monotonically decreasing (increasing), the system is stable. Energy function Energy function was defined as In bipolar notation the complement of vector v is –v The memory transition may end up to v as easily as -v The similarity between initial output vector and v and -v determines the convergence. It has been shown that synchronous state updating algorithm may yield persisting cycle states consisting of two complimentary patterns Example Memorized patterns Hopfield network as a content-addressable memory, we know a priori the fixed points of the network in that they correspond to the patterns to be stored. Fixed point attractor are the minima of the energy function  patterns  An energy contour map for a two-neuron, two-stable-state system  The attractors of the dynamical system are the local minima of the energy function Update (asynchronous updating ) The selection of a neuron to perform the updating is done randomly. The asynchronous (serial) updating procedure described here is continued until there are no further changes to report. In matrix form: Asynchronous Update The output is updated asynchronously. This means that for a given time, only a single neuron (only one entry in vector V ) is allowed to update its output Example: In this example output vector is started with initial value V0, they are updated by m, p and q respectively: