You need to define what you mean by similarity, as the vectors are in spaces of differing dimensions. How will you exploit "they are similar" or "they are dissimilar" ?

My only contribution, no time to discuss more.

Anis,

The definition of similarity is crucial, as Alain wrote.

I would like to add a couple of  points:

Are you working on something on binary messages or codes ?

-Sanjay

Please see if this helps.

1. I think  intuitive concept of similarity of two vectors is given by dot product(inner product) i.e

cos(theta)= A.dot.B/(|A|.|B|) =( at.b / (at.a )(bt.b)

2. Embed lower dim vector in higher dimension e.g

        A = [0,1]  = {0 1  0 0 ]       B= [0 1 0 1 ]          then

     cos(theta) = 1/sqrt(2) = .707

Cheers

Thanks, Anis.

I was waiting for something like this. So putting it in analytical terms:

• You have a set of variables (one associated with each webpage in the example), each of which can assume binary values (0/1).
• Any distance function will accordingly have positive integer values, depending on how many 0's and how many 1's are there !
• Now you want to define similarity based on some functional of the behaviour vector, that should make sense ! Am I right ??

The problem in such information scenarios is that we don't know what is the maximum dimension of the array. In the example for instance, one user may spend the entire T on one webpage, and one unhappy user may open twenty webpages, and not use any of them properly !

I have a suggestion (maybe not a perfect one, but you may wish to consider it !):

Try the functional = (No. of pages opened) / T

This will get rid of your vector dimension problem, and would place all the users on a comparable platform. The formal definition of this could be something like:

Time average (or temporal average) of the L1 norm of the user vector.

It is possible to refine this idea by weighting pages that "were opened earlier" with small weights and the ones that were "opened recently" with higher weights, but that  is simply a matter of detail, I think !

Hope the idea was not too "wild" :-) :-)

Cheers !

-Sanjay

Dear Sanjoy,

Thank you very much for your quick reply.

I am very grateful for all these explanations !

Best regards

Anis

Glad that helped !

Unfortunately Anis, the L2 norm is so widely used all around us, that on many occasions we tend to overlook that the L1 and Linf norms actually take care of a lot of problems !!

Here was an example !!

Cheers...

-Sanjay

1. It seems  akin to classification and Naive Bayesian classification could perhaps  be used..It would be interesting to compare if at all feasible with proposed method.

2. L1 norm as a " measure"  that is suggested has found good results in compression sensing..see e.g     http://statweb.stanford.edu/~candes/

Good luck in your research..

Cheers