For instance, you can initialize it with the space spanned by the first two eigenvectors of PCA. Then you combine the best of two world: starting with a globally optimal embedding, you address the problem of preserving local topology. The problem, of course, is that you need to do the PCA first, which can be computationally prohibitive. Most SOM implementations have an option for PCA initialization (e.g., Somoclu's Python interface and SomPY).

If you have a non-emergent map, that is, you have fewer neurons than data points, then you can randomly sample your data points as initial weight vectors. This has the advantage of being computationally inexpensive.

https://somoclu.readthedocs.org/en/stable/example.html#initialization-with-principal-component-analysis-and-clustering-the-results

https://github.com/sevamoo/SOMPY

The original self-organizing map also included random initializations with a low (close to 0) value and using k-means for initializing vectors. both are probably simple to implement and usually work well.

Dear Noha,

You question is very interesting. The initialization of SOM with non random values depend on the problem itself. In general any process that initialize the weights of SOM depends on another process which uses random initialization such as GA, SA.  But, at least  SOM will start with a global optimal solution.

At the end you should consider the efficiency of the system with respect to time and in that case I would prefer to use random weight initialization from the data set (e.g gray level values of an image divided by the maximum value 200/255; 100/255...etc).

It is also better to test different initialization strategies and all depends on the problem.   

 One more issue that you should worry about using PCA for dimensionality reduction is that it is affected by noise. So, it is advisable not to use it with noisy data. Good luck