4 research outputs found
Complexity of linear circuits and geometry
We use algebraic geometry to study matrix rigidity, and more generally, the
complexity of computing a matrix-vector product, continuing a study initiated
by Kumar, et. al. We (i) exhibit many non-obvious equations testing for
(border) rigidity, (ii) compute degrees of varieties associated to rigidity,
(iii) describe algebraic varieties associated to families of matrices that are
expected to have super-linear rigidity, and (iv) prove results about the ideals
and degrees of cones that are of interest in their own right.Comment: 29 pages, final version to appear in FOC
Degrees of projections of rank loci
We provide formulas for the degrees of the projections of the locus of square
matrices with given rank from linear spaces spanned by a choice of matrix
entries. The motivation for these computations stem from applications to
`matrix rigidity'; we also view them as an excellent source of examples to test
methods in intersection theory, particularly computations of Segre classes. Our
results are generally expressed in terms of intersection numbers in
Grassmannians, which can be computed explicitly in many cases. We observe that,
surprisingly (to us), these degrees appear to match the numbers of Kekul\'e
structures of certain `benzenoid hydrocarbons', and arise in many other
contexts with no apparent direct connection to the enumerative geometry of rank
conditions