SVD to fill in missing values of a matrix

Hi,

I currently have a user by user matrix, with each entry indicating the similarity between the corresponding users. This matrix is sparse, as entries are calculated only for friend pairs.

Now, I want to predict the possible similarity scores between non-friend pairs i.e fill the empty values in the matrix.

I read about latent factor models and the use of SVD to minimise the squared reconstruction error.

But there are a few differences with my use case. I need to perform svd on a user-user matrix, not a user-item matrix. This would mean that I need my result to be of the form A= U U^T , where U is the latent factor matrix for a user. This does not seem to map to SVD, but to Cholesky decomposition. Using Cholesky decomposition does not let me choose the number of latent factors.

I am not sure if I am thinking in the right direction or if there is some fundamental flaw in my approach. Any kind of advice is appreciated.

Thanks.