# Non-linear projection in self organizing maps

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?

• This doesn't really belong to CV.SE either, I think. – Memming Sep 7 '13 at 17:11

Your 3 by 3 SOM is approximating a 2D manifold. Your points mapped to a $3 \times 3$ grid is the "projection". If you had more data, and used a larger grid, say of $100 \times 100$, then it may be more intuitive.
If you want to find a 3D manifold, you could use a $3 \times 3 \times 3$ grid. Is it more clear now?