Algebraic Geometry in an applied setting?

Question

I just saw this paper recently which mentioned that the optimization on a Grassmanian Manifold can be used to get an achieve an best approximation of a multilinear rank of a tensor (in the sense of a multidimensional matrix, also called a hypermatrix). Does anyone happen to know what Grassmanians have to do with tensors and their analysis? And where could I learn more about Grassmanian manifolds (enough to understand the use here, more is of course welcome =)

The paper is here for those interested Paper

(I hope this isn't too specific)

Answer 1

You might also want to check out this paper by Landsberg and Teitler on getting bounds for the Waring rank (and border rank) of symmetric tensors using geometry.

The point here is that the projective bundle on a vector space captures the geometry of its symmetric tensors so one can reformulate questions about say the rank into questions about the geometry of secant varieties on the projective bundle of interest. It turns out one can prove some new bounds using some very classical algebraic geometry.

Answer 2

Well, as Grassmannians parameterize vector subspaces of a vector space, they are pretty much in control of any sort of linear thing, like tensors, so at the least, I don't think it's surprising that they come up. As far as sources on the Grassmannians, I wrote a series of posts on the cohomology of Grassmannians at Rigorous Trivialities, and for a more thorough exposition on some of the same stuff, you can check out the last section of Fulton's "Young Tableux." As far as more analytic properties of Grassmannians, I don't have good references, and most of the time algebraic geometers seem to be interested in cohomology (of various sorts) of flag varieties and of using them to make other moduli spaces. Hope this helps.