Showing papers by "Joseph M. Landsberg published in 2009"

An overview of mathematical issues arising in the Geometric complexity theory approach to VP v.s. VNP

Abstract: We discuss the geometry of orbit closures and the asymptotic behavior of Kronecker coefficients in the context of the Geometric Complexity Theory program to prove a variant of Valiant's algebraic analog of the P not equal to NP conjecture. We also describe the precise separation of complexity classes that their program proposes to demonstrate.

On the ranks and border ranks of symmetric tensors

Abstract: Motivated by questions arising in signal processing, computational complexity, and other areas, we study the ranks and border ranks of symmetric tensors using geometric methods. We provide improved lower bounds for the rank of a symmetric tensor (i.e., a homogeneous polynomial) obtained by considering the singularities of the hypersurface defined by the polynomial. We obtain normal forms for polynomials of border rank up to five, and compute or bound the ranks of several classes of polynomials, including monomials, the determinant, and the permanent.

Kruskal's theorem

Abstract: This is just a short proof of Kruskal's theorem regarding uniqueness of expressions for tensors, phrased in geometric language.

Fubini's theorem in codimension two

Abstract: We classify codimension two analytic submanifolds of projective space X ⊂ CP having the property that any line through a general point x ∈ X having contact to order two with X at x automatically has contact to order three. We give applications to the study of the Debarre-de Jong conjecture and of varieties whose Fano variety of lines has dimension 2n − 4.

Holographic algorithms without matchgates

Abstract: The theory of holographic algorithms, which are polynomial time algorithms for certain combinatorial counting problems, yields insight into the hierarchy of complexity classes. In particular, the theory produces algebraic tests for a problem to be in the class P. In this article we streamline the implementation of holographic algorithms by eliminating one of the steps in the construction procedure, and generalize their applicability to new signatures. Instead of matchgates, which are weighted graph fragments that replace vertices of a natural bipartite graph G associated to a problem P, our approach uses only only a natural number-of-edges by number-of-edges matrix associated to G. An easy-to-compute multiple of its Pfaffian is the number of solutions to the counting problem. This simplification improves our understanding of the applicability of holographic algorithms, indicates a more geometric approach to complexity classes, and facilitates practical implementations. The generalized applicability arises because our approach allows for new algebraic tests that are different from the "Grassmann-Plucker identities" used up until now. Natural problems treatable by these new methods have been previously considered in a different context, and we present one such example.

P versus NP and geometry

Abstract: I describe three geometric approaches to resolving variants of P v. NP, present several results that illustrate the role of group actions in complexity theory, and make a first step towards completely geometric definitions of complexity classes.