Self-organizing map

A self-organizing map (SOM) or self-organizing feature map (SOFM) is a type of artificial neural network that is trained using unsupervised learning to produce a low-dimensional (typically two-dimensional), discretized representation of the input space of the training samples, called a map. Self-organizing maps are different than other artificial neural networks in the sense that they use a neighborhood function to preserve the topological properties of the input space.

This makes SOM useful for visualizing low-dimensional views of high-dimensional data, akin to multidimensional scaling. The model was first described as an artificial neural network by the Finnish professor Teuvo Kohonen, and is sometimes called a Kohonen map.

Like most artificial neural networks, SOMs operate in two modes: training and mapping. Training builds the map using input examples. It is a competitive process, also called vector quantization. Mapping automatically classifies a new input vector.

A self-organizing map consists of components called nodes or neurons. Associated with each node is a weight vector of the same dimension as the input data vectors and a position in the map space. The usual arrangement of nodes is a regular spacing in a hexagonal or rectangular grid. The self-organizing map describes a mapping from a higher dimensional input space to a lower dimensional map space. The procedure for placing a vector from data space onto the map is to find the node with the closest weight vector to the vector taken from data space and to assign the map coordinates of this node to our vector.

While it is typical to consider this type of network structure as related to feedforward networks where the nodes are visualized as being attached, this type of architecture is fundamentally different in arrangement and motivation.

Useful extensions include using toroidal grids where opposite edges are connected and using large numbers of nodes. It has been shown that while self-organizing maps with a small number of nodes behave in a way that is similar to K-means, larger self-organizing maps rearrange data in a way that is fundamentally topological in character.

It is also common to use the U-matrix. The U-matrix value of a particular node is the average distance between the node and its closest neighbors. In a rectangular grid for instance, we might consider the closest 4 or 8 nodes.

Large SOMs display properties which are emergent. Therefore, large maps are preferable to smaller ones. In maps consisting of thousands of nodes, it is possible to perform cluster operations on the map itself.

Learning algorithm
The goal of learning in the self-organizing map is to cause different parts of the network to respond similarly to certain input patterns. This is partly motivated by how visual, auditory or other sensory information is handled in separate parts of the cerebral cortex in the human brain.

The weights of the neurons are initialized either to small random values or sampled evenly from the subspace spanned by the two largest principal component eigenvectors. With the latter alternative, learning is much faster because the initial weights already give good approximation of SOM weights.

The network must be fed a large number of example vectors that represent, as close as possible, the kinds of vectors expected during mapping. The examples are usually administered several times.

The training utilizes competitive learning. When a training example is fed to the network, its Euclidean distance to all weight vectors is computed. The neuron with weight vector most similar to the input is called the best matching unit (BMU). The weights of the BMU and neurons close to it in the SOM lattice are adjusted towards the input vector. The magnitude of the change decreases with time and with distance from the BMU. The update formula for a neuron with weight vector Wv(t) is


 * $$\mathbf{Wv}(t+1)= \mathbf{Wv}(t)+ \Theta(v,t)\alpha(t)(D(t)- \mathbf{Wv}(t))\,$$

where α(t) is a monotonically decreasing learning coefficient and D(t) is the input vector. The neighborhood function Θ (v, t) depends on the lattice distance between the BMU and neuron v. In the simplest form it is one for all neurons close enough to BMU and zero for others, but a gaussian function is a common choice, too. Regardless of the functional form, the neighborhood function shrinks with time. At the beginning when the neighborhood is broad, the self-organizing takes place on the global scale. When the neighborhood has shrunk to just a couple of neurons the weights are converging to local estimates.

This process is repeated for each input vector for a (usually large) number of cycles λ. The network winds up associating output nodes with groups or patterns in the input data set. If these patterns can be named, the names can be attached to the associated nodes in the trained net.

During mapping, there will be one single winning neuron: the neuron whose weight vector lies closest to the input vector. This can be simply determined by calculating the Euclidean distance between input vector and weight vector.

While representing input data as vectors has been emphasized in this article, it should be noted that any kind of object which can be represented digitally and which has an appropriate distance measure associated with it and in which the necessary operations for training are possible can be used to construct a self-organizing map. This includes matrices, continuous functions or even other self-organizing maps.

Preliminary definitions
Consider a 10×10 array of nodes each of which contains a weight vector and is aware of its location in the array. Each weight vector is of the same dimension as the node's input vector. The weights are initially set to random values.

Now we need input to feed the map. (The generated map and the given input exist in separate subspaces.) We will create three vectors to represent colors. Colors can be represented by their red, green, and blue components. Consequently our input vectors will have three components, each corresponding to a color space. The input vectors will be: R = <255, 0, 0> G = <0, 255, 0> B = <0, 0, 255>

Variables
Vectors are in bold t = current iteration λ = limit on time iteration Wv = current weight vector D = target input Θ(t) = restraint due to distance from BMU - usually called the neighbourhood function α(t) = learning restraint due to time

Stepping through the algorithm

 * 1) Randomize the map's nodes' weight vectors
 * 2) Grab an input vector
 * 3) Traverse each node in the map
 * 4) Use Euclidean distance formula to find similarity between the input vector and the map's node's weight vector
 * 5) Track the node that produces the smallest distance (this node is the best matching unit, BMU)
 * 6) Update the nodes in the neighbourhood of BMU by pulling them closer to the input vector
 * 7) $$\mathbf{Wv}(t+1)= \mathbf{Wv}(t)+ \Theta(v,t)\alpha(t)(D(t)- \mathbf{Wv}(t))\,$$
 * 8) Increment t and repeat from 2 while t < λ

Interpretation
There are two ways to interpret a SOM. Because in the training phase weights of the whole neighborhood are moved in the same direction, similar items tend to excite adjacent neurons. Therefore, SOM forms a semantic map where similar samples are mapped close together and dissimilar apart.

The other way is to think of neuronal weights as pointers to the input space. They form a discrete approximation of the distribution of training samples. More neurons point to regions with high training sample concentration and fewer where the samples are scarce.

SOM may be considered a nonlinear generalization of Principal components analysis (PCA). It has been shown, using both artificial and real geophysical data, that SOM has many advantages over the conventional feature extraction methods such as Empirical Orthogonal Functions (EOF) or PCA.

Alternatives
Generative topographic maps (GTM) are a potential alternative to SOMs. In the sense that GTM explicitly requires a smooth and continuous mapping from the input space to the map space, it is topology preserving. However, in a practical sense, this measure of topological preservation is lacking.

Implementations

 * Orange, a free data mining software suite, module orngSOM