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]
