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] >
