A simple example is explored in attached pdf by building a matrix a with given user rating and movie claasification and weights , but we do not seem to recover our original data components , hence some exploration was done using Scilab (in pdf). Would welcome constructive insights and comments .
Good morning, I have studied your PDF and let me allow following comments.
1. You construct the final matrix as the sum of two matrices, these partial matrices are NOT orthogonal. Consequently, you can NOT recover these partial matrices exactly from the SVD of the final matrix, because in the SVD ALL matrices of type $\sigma_i*u_i*v_i^t $ are orthogonal!
2. There are formal errors in your PDF file - for example, the element $la(3,3)$ does not exist.
Dear Gabriel Oksa
Thank you for your helpful pointers and patience.. Yes all the vectors in U and V in svd are orthognal. I was experimenting with the idea that we could take separate rating data and combine them with assumed weightings. From your useful observation it seems we must only add orthogonal new information as new component matrix and correct old component matrix with some projection from new data. This will help in continuous useful updation I think.
Yes la( 3,3) is not in the code..I just assumed it as zero because using eigen values of a*a' we get:
d=clean(spec(a*a'))
d = 0.
20.197218
2737.8028
sqrt(d)
ans = 0.
4.4941315
52.324017
Cheers
Now you have that Sigma is supposed to indicate weight of classification, and U the user preference , while V is the set movie groups. Since your synthetic U and V are not orthonormal, it is not a correct version of SVD.
Two options.
One is whitening your data, so that U and V would be orthonormal matrices.
Another option might be adding a more time-index. Say, Monday, Tuesday, Wednesday, Thursday, Friday, Saturday, Sunday. Then instead of having a matrix U,V,Sigma, you will have another time-matrix W. Then your original data matrix X will be changed as the tensor data T, which is T=sum_i sigma(i) u(i) o v(i) o w(i), with o as the tensor product. This setup has uniqueness presentation, i.e., given T, you can uniquely recover U,V,W,Sigma without the orthonormal conditions on U,V,W. In literature, this is called collaborative filtering.
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