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From Sean Owen <sro...@gmail.com>
Subject Re: Singular vectors of a recommendation Item-Item space
Date Thu, 25 Aug 2011 21:50:21 GMT
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 <dscheffy@gmail.com> 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 =)
>

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