I've been playing around with this problem for the last week or so (or at least this problem as I understood it based on your initial commentary Lance) -- but purely in R using smaller data so I can 1. get my head wrapped around the problem, and 2. get more familiar with R.
To make the problem a little more tenable I limited my sample to 200 movies and 10,000 users (taking the most rated movies from 2004 and 2005 based on NF's dataset -- I know, I should really switch back to the grouplens dataset...) I'm also only looking at binary data at the moment -- I treat any rating above 3 as a movie you liked and anything 3 or below as the same as not having rated the movie. So I take this 200 x 10,000 matrix of 1s and 0s and I run a truncated SVD on it so that I can project it onto a 10 dimensional space. M<-initial data s_m<- svd(M,10,10) U<-s_m$u S<-diag(s_m$d[1:10]) V<-s_m$v So U is a 200 row by 10 column matrix -- each row represents the eigenvector of a given movie, and each column represents one Lance's so called axes of interest. So what I did next was spit out the top and bottom n movie titles for each of these 10 dimensions. I found it was important to show more than one movie title for each side of the dimensions, otherwise the results might be somewhat misleading. I then went through the 10 dimensions and qualitatively answered for myself whether I was strongly or weakly aligned in one direction, or not aligned in anyway on this dimension. Personally I usually found I only felt strongly aligned on 2 of the ten, and weakly aligned on another 2. I then normalized U across each of the ten dimensions and for each movie added up it's z score in that dimension by my alignment in that dimension. I then sorted the results and displayed the movie titles -- it was a pretty accurate ranking of movies as I like them. scaled <- apply(U,2,scale) me <- c(0,2,1,0,-1,1,0,0,0,0) dim(me) <- c(10,1) recommendations <- scaled %*% me I imagine few users would want to bother, but I can see where it would be a relatively quick way to train a recommender. Here's the problem though -- I can get it to work using the method I've described above, but I can't quite figure out how to use it to generate an eigenvector for the user. For existing users I can always generate predictions by matrix multiplying U %*% S %*% t(V)[,user] and then sorting by the results. It would be nice to use a consistent model. I can't quite see the math to generate an equivalent equation though. On Wed, Aug 17, 2011 at 3:52 AM, Lance Norskog <[email protected]> wrote: > Sharpened: > > > http://ultrawhizbang.blogspot.com/2011/08/singular-vectors-for-recommendations.html > > On Wed, Aug 10, 2011 at 11:53 PM, Sean Owen <[email protected]> wrote: > > You may need to sharpen your terms / problem statement here : > > > > What is a geometric value -- just mean a continuous real value? > > So these are item-feature vectors? > > > > The middle bit of the output of an SVD is not a singular vector -- it's a > > diagonal matrix containing singular values on the diagonal. > > The left matrix contains singular vectors, which are not eigenvectors > except > > in very specific cases of the original matrix. > > > > Singular vectors are the columns of the left matrix, not rows, whereas > items > > corresponds to its rows. What do you mean about relating them? > > What do you mean by the "hot spot" you are trying to find? > > A vector does not express two end-points, no. You could think of (X,Y) as > > corresponding to a point in 2-space, or could think of it as a ray from > > (0,0) to (X,Y), but you could think of it as (100,200) to (100+X,200+Y) > just > > as well. There are not two point implied by anything here. > > > > > > How do you get points from the original item-feature space into the > > transformed, reduced space? While I think this is an imprecise answer: if > A > > = U Sigma V^T then you can think of (Sigma V^T) as like the > change-of-basis > > transformation that does this. > > > > > > On Wed, Aug 10, 2011 at 10:54 AM, Lance Norskog <[email protected]> > wrote: > > > >> Zeroing in on the topic: > >> > >> I have: > >> 1) a set of raw input vectors of a given length, one for each item. > >> Each value in the vectors are geometric, not bag-of-words or other. > >> The matrix is [# items , # dimensions]. > >> 2) An SVD of same: > >> left matrix of [ # items, #d features per item] * singular > >> vector[# features] * right matrix of [#dimensions features per > >> dimension, #dimensions]. > >> 3) The first few columns of the left matrix are interesting singular > >> eigenvectors. > >> > >> I would like to: > >> 1) relate the singular vectors to the item vectors, such that they > >> create points in the "hot spots" of the item vectors. > >> 2) find the inverses: a singular vector has two endpoints, and both > >> represent "hot spots" in the item space. > >> > >> Given the first 3 singular vectors, there are 6 "hot spots" in the > >> item vectors, one for each end of the vector. What transforms are > >> needed to get the item vectors and the singular vector endpoints in > >> the same space? I'm not finding the exact sequence. > >> > >> A use case for this is a new user. It gives a quick assessment by > >> asking where the user is on the few common axes of items: > >> "Transformers 3: The Stupiding" v.s. "Crazy Bride Wedding Love > >> Planner"? > >> > >> -- > >> Lance Norskog > >> [email protected] > >> > > > > > > -- > Lance Norskog > [email protected] >
