Difficulty in understanding a vector quantization algorithm Neural gas for vector quantizationpaper explains a technique for symbolizing or quantizing data. Algorithm presents the algorithm in Section 4. An application in EEG data symbolization is presented Application. In the Application it is shown that an n- dimensional data like EEG recordings is vectorized to 1D. But Neural gas is basically an unsupervised clustering algorithm where data can be assigned to more than one cluster. 
The definition of Vector Quantization (VQ) is the process of quantizing
n-dimensional input vectors to a limited set of
n-dimensional output vectors referred to as code-vectors. The set of possible code-vectors is called the codebook. The codebook is
usually generated by clustering a given set of training vectors (called training set),
the codebook is then used to quantize input vectors. 
What I do not understand is that how a multi dimensional data is converted to single dimension as in the Application paper and the module where the symbols are assigned. In the Algorithm, I did not understand what is "reference vector $w_{c_i}$, the set $A$ and Step 3 where it says to order all elements of $A$ according to distance. What is the distance measure here?
Can somebody please explain the algorithm in simple terms how the symbolization is done? Thank you
 A: Neural Gas is just a soft version of k-means clustering.
in k-means , you just move the closest reference vector towards your training pattern, in neural gas, you are also moving the other reference vectors ( weighted according to their "closeness" , ie 1 -nearest, ...,k kth-nearest refence vector).  
the point is that this soft version can [potentially/and more slowly] find better clusters because it avoids getting stuck in local minima... so after training you are just using the nearest reference vector to symbolise the data
reference vector w_{c_i} is then one of the reference vectors (code-vectors), in order of closeness ( just Euclidean distance) to the current training vector.
so "where it says to order all elements of A according to distance. What is the distance measure here?" -. A is just your codebook (of refence vectors). for each training pattern you rank the refence vectors according to how close they are to that training pattern, and apply the training rule.  the distance measure is just euclidean distance.
symbolisation is done by just returning the index into your codebook of vectors... it is not  turning it into a 1 dimensional space . its just like reading off the position in a dictionary - the position does not encode any "useful" information (ie about the meaning off the word)
