advertisement: compare things at compare-stuff.com!

Self-organising maps - Overview
and Methods

Kohonen's self-organising mapkohonen:som1,kohonen:som2 is an unsupervised learning procedure based on a model of laterally interacting neuronal units, resembling mechanisms proposed for stimulus recognition in the cerebral cortex[Miller, 1994,Kohonen, 1993,Durbin & Mitchison, 1990]. In broad overview, it projects high dimensional input data onto an ordered two-dimensional grid, known as the map. As shown in Figure 5.1, the map grid consists of reference vectors of the same dimensionality as the input data. The mapping process is iterative, non-deterministic and mathematically uncharacterised, but has the effect of preserving local ordering relationships (as can be seen in the figure). There follows a more rigorous explanation.

The map is a $M \times N$ array of reference vectors $r_{m,n}$. At time $t=0$, the map is initialised with random vectors within the ranges of the input vectors. A total of $I$ training vectors $v_i$ are inputed to Equations 5.1 and 5.2 at time points $t=[0..T]$ (input vectors repeated as necessary):

$$\vert\vert v_i - r_w \vert\vert = \min ( \vert\vert v_i - r_{m,n} \vert\vert ) \;\forall m,n$$ (9)

$$r_{m,n}(t+1) = r_{m,n}(t) + k(m,n,r_w,d,t)\,a[1 - t/T]\,[v_i - r_{m,n}(t)] \;\forall m,n$$ (10)

where $r_w$ is the `winning' reference vector, the closest reference vector to the input vector using the Euclidean distance metric. The winning reference vector and its neighbours defined by the neighbourhood kernel function $k(m,n,r_w,d,t)$ are updated towards the input vector (Equation 5.2). The kernel function basically defines a set of neighbouring cells (shown in grey in Figure 5.1). The kernel radius and learning rate, initially set to $d$ and $a$ respectively, decrease linearly with time to zero at the end of the mapping procedure. At the end of the learning phase the training vectors (or any other vector of the same dimension) can be mapped to a winning vector on the output grid using Equation 5.1.

Figure 5.1: Some basic principles and definitions for the Kohonen self-organising map.