If you take the item vector an existing user, multiply that by the left-hand SVD matrix, and multthe resulting vector[i] * 1/singularvalues[i], you should get the item's row in the left-hand column. So, the left-hand column times 1/singular values gives you the projection for a new user's item vector >into this space of users<. Which gives you a user-user recommender for new users.
On Thu, Aug 25, 2011 at 2:50 PM, Sean Owen <[email protected]> wrote: > The 200x10 matrix is indeed a matrix of 10 singular vectors, which are > eigenvectors of AA'. It's the columns, not rows, that are > eigenvectors. > > The rows do mean something. I think it's fair to interpret the 10 > singular values / vectors as corresponding to some underlying features > of tastes. The rows say how much each user expresses those 10 tastes. > The matrix of right singular vectors on the other side tells you the > same thing about items. The diagonal matrix of singular values in the > middle also comes into play -- it's like a set of multipliers that say > how important each feature is. (This is why we cut out the singular > vectors / values that have the smallest singular values; it's like > removing the least-important features.) So really you'd have to stick > those values somewhere; Ted says it's conventional to put "half" of > each (their square roots) with each side if anything. > > I don't have as good a grasp on an intuition for the columns as > eigenvectors. They're also a set of basis vectors, and I had > understood them as like the "real" bases of the reduced feature space > expressed in user-item space. But I'd have to go back and think that > intuition through again since I can't really justify it from scratch > again in my head just now. > > > On Thu, Aug 25, 2011 at 10:21 PM, Jeff Hansen <[email protected]> wrote: >> Well, I think my problem may have had more to do with what I was calling the >> eigenvector... I was referring to the rows rather than the columns of U and >> V. While the columns may be characteristic of the overall matrix, the rows >> are characteristic of the user or item (in that they are a rank reduced >> representation of that person or thing). I guess you could say I just had to >> tilt my head to the side and change my perspective 90 degrees =) >> > -- Lance Norskog [email protected]
