SOM and Adaptive Coordinates

The Adaptive Coordinates (AC) approach is an extension to the standard learning procedure for Kohonen's self-organizing map (SOM) [2]. The standard training procedure itself remains unmodified as follows: Input signals $x \in \Re^{n}$ are presented to the map, consisting of a grid of units with n-dimensional weight vectors, in random order. An activation function based on some metric (e.g. the Euclidean Distance) is used to determine the winning unit (the 'winner'). In the next step the weight vector of the winner as well as the weight vectors of the neighboring units are modified following some learning rate in order to represent the presented input signal more closely.

The basic idea of the AC approach is to visualize clusters in input space in a 2-dimensional output space. By extending the basic training procedure we try to mirror the movement of the nodes' weight vectors in the high-dimensional input space in a 2-dimensional output space. For mirroring this movement, each unit i is assigned a position in this output space, with its position as given by two adaptive coordinates $\langle ax_i, ay_i \rangle$ initially being identical to the unit's position in the map grid. During the learning process, at each step t the distances between each unit and the presented input signal are stored in a table Dist(t). After adapting the weight vectors following the standard training rule, this distance table is calculated again for the same input signal as Dist(t+1). Based on these tables the relative change in distance between the weight vector of every unit i and the presented input signal can be calculated as

$$\Delta Dist_{i}(t+1) = \frac{Dist_{i}(t) - Dist_{i}(t+1)}{Dist_{i}(t)}$$ (2)

describing the movement of the unit's weight vectors towards the presented input signal in input space. This movement is now performed analogously in the 2-dimensional output space with the winning unit being representative of the presented input signal, i.e. the adaptive coordinates of every unit but the winner are modified such that the unit's new position is moved by the same fraction as given in $\Delta Dist$ towards the winner's position given by $\langle ax_c, ay_c \rangle$. Thus, the movement of the adaptive coordinate ax_i can be given as

$$ax_i(t+1) = ax_i(t) + \Delta Dist_i(t+1) \cdot (ax_c(t) - ax_i(t))$$ (3)

with the adaption of ay_i being performed analogously. Using the adaptive coordinates <ax_i, ay_i> to plot the units location in the 2-dimensional output space allows the visualization of the clustering learned by the SOM.