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What are Kohonen Self-Organizing Maps?

Arooj Fatima

The Kohonen Self-Organizing Mapis named after the researcher Dr. Teuvo Kohonen.

A Kohonen Self-Organizing Map is a map that is used for maintaining neighbor topology. It is referred to as a neural network that is trained by competitive learning.

Neighbor topologies

Bellow are the two most used topologies in the Kohonen Self-Organizing Map.

Rectangular grid

The following explains the rectangular grid topology:

  • In the distance of 2 grids: 24 nodes
  • In the distance of 1 grid: 16 nodes
  • In the distance 0 grid: 8 nodes

There is a total difference of 88 nodes between each grid.

Hexagonal topology

The following explains the hexagonal grid topology:

  • In the distance of 2 grids: 18 nodes
  • In the distance of 1 grid: 12 nodes
  • In the distance 0 grid: 6nodes

There is a total difference of 8 nodes between each grid.

Each node has its own coordinates after clustering, which can be calculated with the Pythagorean theorem using the Euclidean distance between the two nodes.

Structure

A Kohonen Self-Organizing Map is different from common artificial neural networks in its:

  • architecture properties
  • algorithmic properties

A Kohonen Self-Organizing Map consists of a single layer linear 2D grid of neurons. The nodes do not know the values of their neighbors. The weights of links are updated as a function of the given inputs. However, all the nodes on the grid are directly linked to the input vectors.

Algorithm

Variables used in the algorithm

  • x = input vector
  • x(t) =input vector’s occurrence at iteration t
  • n = total number of iterations a network can perform
  • t = current iteration
  • i = row coordinate
  • j = column coordinate
  • λ = time constant
  • d = distance between a node and the BMU
  • w = weight vector
  • w_ij = weight of the links between the nodes i,j
  • β_ij = neighborhood function, representing and decreasing a node i, j distance from the BMU
  • σ(t) = radius of the neighborhood function, it decides how far neighbor nodes are inspected in the 2D grid when updating vectors. It decreases over time.

Steps of algorithm

Step 1

  • Every node weight is initialized to a random value.

Step 2

  • Select a random input vector.

Step 3

  • Determine the Euclidean distance between the weight vector and the input vector connected with the first node considering t, i, j =0.

Step 4

  • Note the node that generates the smallest distance.

Step 5

  • Repeat steps 3 and 4 for all nodes.

Step 6

  • Determine the complete BMU, i.e., node with the smallest distance from each already determined node.

Step 7

  • Calculate βij(t) its radius σ(t) of BMU in Kohonen Self-Organizing Map.

Step 8

  • Repeat for all nodes in the BMU neighborhood.

Step 9

  • Update the weight vector of the first node in the neighborhood of the BMU by including a fraction of the difference between the input vector and the weight of the neuron.

Step 10

  • Repeat the iterations until the total number of iterations are completed.

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