For simplicity let's consider a brand-new user first, not a new rating for existing user. I'll use the notation from my slides that you mention, A = X * Y'. To clarify, I think you mean you have a new A_u row, and want to know X_u.
The two expressions are not alternatives, they're the same thing, though abusing notation a little. In the first that's not a real inverse since Y' is not even square. It needs to be something that works as a right-inverse. The second elaborates what that is -- that is an inverse, of a square matrix. As you might have seen on this list I actually investigated this part a lot last week since I thought there was a numerical problem. There is a gotcha here, but it's not to do with inverses. Yes you can use a QR decomposition to find the inverse. It's fast. There is a QR decomposition in the project, or, I've had good luck with Commons Math. (In fact 3.2 has a proper rank-revealing QR decomposition now, which is useful for other reasons in this context.) I ended up not actually computing the inverse, and instead using a QR decomposition to solve A_u = X_u * Y' for X_u. It's nearly exactly the same thing, but more principled. Back to handling a new event for an existing user -- it gets trickier here since you don't want to fold-in A_u = [ .. 0 0 1 0 0 .. ] -- it's too big, especially if A_u already has a value near 1 in that spot. Note that you also want to consider updating Y similarly; you've learned about the user but you also learned about the item. Same thing, just for the other matrix. Sean On Wed, Apr 10, 2013 at 9:29 PM, Chloe <chloe.gu...@gmail.com> wrote: > Hi everyone, > > I am reaching out to the list requesting some help/advice on implementing > fold-in with the Alternating Least Squares algo in Mahout, a problem on > which I am stumped. I've read other posts on the list and over on SO, like: > > http://stackoverflow.com/questions/12444231/svd-matrix-conditioning- > how-to-project-from-original-space-to-conditioned-spac > and > http://stackoverflow.com/questions/12857693/mahout-how-to-make- > recommendations-for-new-users > including the slide show talk posted in the last link. This slide show seems > to be the most relevant to my purposes, specifically slide 14, but I can't > fully > understand it. > > At the end of ALS I have Ratings = A = UM', where U is user feature matrix > and M is item feature matrix. > For an existing user with a new rating, I plan to update the Auser only,which > is (1 x number of items) by replacing/adding new rating at the appropriate > index and then do: > Uuser = Auser * V' ( 1 x items)(items x features) > Sub Uuser back into U and get new recommendations > > How do I address a NEW user? From the slide: > X = A (Y')^-1 > or > X = A Y ((Y'Y) ^-1) > > Is the 2nd preferable? If so, why? I would appreciate a more 'wordy' > explanation or someone pointing me to any papers I can read on this. Also, > as a slight offshoot, given that mahout doesn't have matrix inversion, is it > more advisable to use QR decomp to find a pseudoinverse or an outside > package to do inversion? > > Apologies for the long post which I did try and make concise, and thank > you for any help you offer. > > Chloe > >
