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Write short notes on the following. i) Applications of Hopfield Network for Travelling sales man problem ii) Associative Memory

i) Applications of Hopfield Network for Travelling Salesman Problem

Overview

The Travelling Salesman Problem (TSP) is a classic combinatorial optimisation problem where the goal is to find the shortest possible route that visits a set of cities exactly once and returns to the origin city. The Hopfield network, a type of recurrent neural network, can be adapted to solve TSP by mapping the problem into a neural network framework.

Key Concepts

  1. Energy Function: The Hopfield network is governed by an energy function that the network seeks to minimise. For TSP, the energy function is designed to represent the total distance traveled.

  2. Neurons and States: In the context of TSP, each neuron represents a city at a specific position in the tour. The state of each neuron indicates whether the city is visited at that position.

Encoding TSP in a Hopfield Network

  1. Neurons Representation:

    • The network consists of ( N^2 ) neurons, where ( N ) is the number of cities. Each neuron ( x_{ij} ) represents city ( i ) at position ( j ) in the tour.

  2. Constraints:

    • City Visit Constraint: Each city must be visited exactly once.
    • Tour Position Constraint: Each position in the tour must be occupied by exactly one city.

  3. Energy Function: The energy function for TSP in a Hopfield network includes terms to enforce the constraints and minimize the total distance: [$$ E = A \sum_{i} \left( \sum_{j} x_{ij} - 1 \right)^2 + B \sum_{j} \left( \sum_{i} x_{ij} - 1 \right)^2 + C \sum_{i} \sum_{j} \sum_{k \neq j} d_{ik} x_{ij} x_{i,k+1} $$] where ( d_{ik} ) is the distance between cities ( i ) and ( k ), and ( A, B, C ) are coefficients that balance the constraints.

Steps to Solve TSP Using Hopfield Network

  1. Initialization: Initialize the states of the neurons randomly.
  2. Update Rule: Update the state of each neuron asynchronously based on the Hopfield update rule: [$$ x_{ij}(t+1) = \text{sgn}\left(\sum_{k,l} W_{ij,kl} x_{kl}(t) - \theta_{ij}\right) $$] where ( W_{ij,kl} ) are the weights and ( \theta_{ij} ) are the biases designed to minimize the energy function.
  3. Iteration: Repeat the updates until the network converges to a stable state representing a valid tour.
  4. Extract Solution: Decode the final state of the network to extract the tour.

Advantages and Limitations

  • Advantages:

    • Provides a parallel and distributed approach to solving TSP.
    • Capable of finding good solutions for small to moderately sized problems.

  • Limitations:

    • Computational complexity increases with the number of cities, making it less practical for large-scale TSP.
    • May converge to local minima, requiring multiple runs or additional heuristics to find optimal solutions.

ii) Associative Memory

Overview

Associative memory in artificial neural networks refers to the capability of the network to store and recall information based on partial or noisy inputs. It mimics the human brain's ability to remember entire experiences or pieces of information when provided with cues.

Key Characteristics

  1. Pattern Storage and Recall: Associative memory systems can store multiple patterns and retrieve the complete patterns from partial or distorted inputs.
  2. Content Addressability: Information is retrieved based on the content of the input rather than specific addresses, allowing for flexible and robust data retrieval.
  3. Distributed Representation: Information is stored across the entire network, enhancing fault tolerance and robustness.
  4. Noise Tolerance: Capable of recalling correct patterns even when the input data is noisy or incomplete.

Types of Associative Memory

  1. Auto-associative Memory: Stores and recalls the same patterns. A classic example is the Hopfield network.
  2. Heteroassociative Memory: Stores pairs of patterns, allowing the recall of one pattern when presented with its associated pattern. Examples include Bidirectional Associative Memory (BAM).

Examples

  1. Hopfield Network:

    • Architecture: Consists of fully connected neurons with symmetric weights.
    • Energy Function: The energy function ( E ) guides the network to stable states corresponding to stored patterns: [$$ E = -\frac{1}{2} \sum_{i \neq j} W_{ij} s_i s_j + \sum_i \theta_i s_i $$]
    • Update Rule: Neurons update asynchronously to minimize the energy: [$$ s_i(t+1) = \text{sgn}\left(\sum_{j \neq i} W_{ij} s_j(t) - \theta_i \right) $$]

  2. Bidirectional Associative Memory (BAM):

    • Architecture: Consists of two layers (input and output) with bidirectional connections.
    • Update Rule: Updates input and output units to reinforce the associations between pattern pairs.

Applications

  1. Pattern Recognition: Recognising patterns from noisy or partial inputs, useful in image and speech recognition.
  2. Data Retrieval: Retrieving relevant records in databases and information systems based on partial queries.
  3. Error Correction: Correcting errors in data transmission by recalling the closest stored pattern.
  4. Cognitive Models: Simulating aspects of human memory and cognition for understanding brain functions.

Conclusion

Associative memory and its applications, such as solving the Travelling Salesman Problem with Hopfield networks, demonstrate the powerful capabilities of neural networks in pattern storage, recall, and optimisation tasks. These networks' ability to store patterns and retrieve them based on partial or noisy inputs makes them valuable tools in various fields, from cognitive modelling to practical engineering solutions.