I have difficulty understanding how self organizing maps (SOM) are doing dimensionality reduction. Can anybody provide a useful explanation to me?
Suppose we have 20 training data points in 50 dimensions. Let's say, I have specified 3 by 3 SOM (lattice with 9 points), I embed my manifold (3 by 3 lattice) to 50-D space and after the training process, each data point is mapped to one of the 9 points (nodes) in my manifold. Now, my embedded manifold (3 by 3 SOM) are 50-D. So how come I'm back to 2-D dimensions? I mean, where is this non-linear projection?