Are you asking what the left and right vectors mean in general in the SVD? S is a re-expression of the original matrix's transformation, but in a different and more natural basis. (Actually it's an approximation, since small singular values are tossed out, and the rank of S is therefore much smaller since those 0s might as well not exist.) So U and V are really change-of-basis transformations -- transforming into S's world of basis vectors and back out again.
In more CF-oriented terms, S is an expression of pseudo-users' preferences for pseudo-items. And then U expresses how much each real user corresponds to each pseudo-user, and likewise for V and items. To put out a speculative analogy -- let's say we're looking at users' preferences for songs. The "pseudo-items" that the SVD comes up with might correspond to something like genres, or logical groupings of songs. "Pseudo-users" are something like types of listeners, perhaps corresponding to demographics. Whereas an entry in the original matrix makes a statement like "Tommy likes the band Filter", an entry in S makes a statement like "Teenage boys in moderately affluent households like industrial metal". And U says how much Tommy is part of this demographic, and V tells how much Filter is industrial metal. (Unfortunately, the SVD doesn't tell you these interpretations, and interpretations of S are rarely so neat as in this made-up analogy.) On Mon, Nov 22, 2010 at 7:52 AM, Lance Norskog <[email protected]> wrote: > This post is inspired by this tutorial, which talks about interpreting > the U and V matrices: > > > http://www.puffinwarellc.com/index.php/news-and-articles/articles/30-singular-value-decomposition-tutorial.html > > Given a DataModel that generates preferences between all Users and all > Items, lets take two Users and three Items: > I I I > U 0.5 0.2 0.1 > U 0.8 0.3 0.2 > > What can we learn from an SVD factorization? > > SVD gives 3 matrices and a scalar: U, a singular value matrix that > signifies the actual rank of the matrix, and transpose(V). For > simplicity, do the 1-dimensional factorization, which gives left and > right vectors instead of scalars. Ignoring the scaling matrix, we get > the Left and Right singular vectors. > > The Left Singular Vector is: (column A x rows U1 and U2) > > ____A__ > U1 > U2 > > The Right Singular Vector is: (row B x columns I1, I2, and I3) > > ___I1___I2___I3 > B > > Now, the question: what do the Left and Right vectors encode? What do > column A and row B mean? > > -- > Lance Norskog > [email protected] >
