The eigenvectors form an orthonormal matrix. Thus R R' = I (the unit, not
your I).
Switching to use u for the user representation:
u R = r
u R R' = r R'
u = r R'
Note that the right eigenvectors are normally written as a pre-transposed
value. Thus the original rating matrix is approximated by:
A = U diag(d) V'
Here your R = V' and R R' = V' V = I is the slightly more traditional way to
express orthonormality.
On Fri, Feb 25, 2011 at 1:06 PM, Chris Schilling <[email protected]> wrote:
> Hey Ted,
>
> This is what I though. So, let's say I have a *new* user that I want to
> estimate the basis vector for a new user using the SVD I already computed.
>
> I end up having something like:
> l*R = r
>
> where l is the user basis I want to calculate and R is the feature basis
> for the items and r is the users ratings. My linear algebra is a bit rusty.
> But this almost looks like least squares. In any case, what is the next
> step. Given the users ratings and the singular vectors for the items, how
> do I estimate the user's singular vector?
>
> Thanks for the help
> Chris
>
>
>
> On Feb 25, 2011, at 10:48 AM, Ted Dunning wrote:
>
>
> Small numbers of new users rarely cause the universe to be all that
> different. That means that you can usually express their behavior pretty
> accurately in terms of the old basis vectors. Expressed mathematically, if
> you have a large number of vectors sampled from some relatively low rank
> sub-space then the basis for that sub-space as estimated from your first
> sample will be pretty good and thus will be a decent basis for another
> tranche of vectors from the same distribution.
>
> So the answer to your question is, yes, there are easy assumptions to
> simplify this process and yes, you can assume negligible effect and just
> rebuild occasionally.
>
> On Fri, Feb 25, 2011 at 10:42 AM, Chris Schilling <[email protected]>wrote:
>
>> Hello,
>>
>> So, I have begun working with the SVDRecommender implementation.
>> Obviously, a matrix factorization technique will not play nice when we try
>> to add an anon/new user (say via the PAUDM) because we have not
>> re-factorized.
>>
>> So my question is more from the linear algebra standpoint: is it possible
>> to estimate the new singular vector when adding a small amount of
>> information (i.e. a new user or a new item) to the matrix? Are there
>> assumptions that can be made to simplify this estimate? For instance, could
>> we assume that the addition of a single row or column will have negligible
>> effect on the already computed singular vectors?
>>
>> If this is possible, I would be interested in playing around with the
>> implementation. Can you point me in the right direction? I'll do some more
>> research over the next few days as well.
>>
>>
>> Finally, I have a question about the sequential factorizations in Mahout.
>> Once I have my original matrix factorized, what is the easiest way to
>> serialize the results of the factorization for reading back in at a later
>> time? Would I loop through all the items and users and dump the features
>> for each? How to read that back in then?
>>
>> Thanks for all the help, Much appreciated,
>> Chris
>>
>>
>
>
