Geometric Complexity Theory and Tensor Rank

Communications on Applied Mathematics and Computation

The purpose of this note is to give a linear algebra algorithm to find out if a rank of a given tensor over a field F is at most k over the algebraic closure of F, where k is a given positive integer. We estimate the arithmetic complexity of our algorithm.

Annali di Matematica Pura ed Applicata (1923 -)

We give a sufficient criterion for a lower bound of the cactus rank of a tensor. Then we refine that criterion in order to be able to give an explicit sufficient condition for a non-redundant decomposition of a tensor to be minimal and unique.

The border rank of the matrix multiplication operator for n by n matrices is a standard measure of its complexity. Using techniques from algebraic geometry and representation theory, we show the border rank is at least 2n^2-n. Our bounds are better than the previous lower bound (due to Lickteig in 1985) of 3/2 n^2+ n/2 -1 for all n>2. The bounds are obtained by finding new equations that bilinear maps of small border rank must satisfy, i.e., new equations for secant varieties of triple Segre products, that matrix multiplication fails to satisfy.

ANNALI SCUOLA NORMALE SUPERIORE - CLASSE DI SCIENZE

In this paper we introduce a new method to produce lower bounds for the Waring rank of symmetric tensors. We also introduce the notion of e-computability and we use it to prove that Strassen's Conjecture holds in infinitely many new cases.

Theoretical Computer Science, 2000

We consider the problem of the presence of short cycles in the graphs of nonzero elements of matrices which have sublinear rank and nonzero entries on the main diagonal, and analyze the connection between these properties and the rigidity of matrices. In particular, we exhibit a family of matrices which shows that sublinear rank does not imply the existence of triangles. This family can also be used to give a constructive bound of the order of k 3=2 on the Ramsey number R(3; k), which matches the best-known bound. On the other hand, we show that sublinear rank implies the existence of 4-cycles. Finally, we prove some partial results towards establishing lower bounds on matrix rigidity and consequently on the size of logarithmic depth arithmetic circuits for computing certain explicit linear transformations.

2021

Abstract. We view the determinant and permanent as functions on directed weighted graphs and introduce their analogues for the undirected graphs. We prove that the task of computing the undirected determinants as well as permanents for planar graphs, whose vertices have degree at most 4, is #P-complete. In the case of planar graphs whose vertices have degree at most 3, the computation of the undirected determinant remains #P-complete while the permanent can be reduced to the FKT algorithm, and therefore is polynomial. The undirected permanent is a Holant problem and its complexity can be deduced from the existing literature. The concept of the undirected determinant is new. Its introduction is motivated by the formal resemblance to the directed determinant, a property that may inspire generalizations of some of the many algorithms which compute the latter. For a sizable class of planar 3-regular graphs, we are able to compute the undirected determinant in polynomial time.

SIAM Journal on Matrix Analysis and Applications

We give sufficient conditions on a symmetric tensor S ∈ S d F n to satisfy the equality: the symmetric rank of S, denoted as srank S, is equal to the rank of S, denoted as rank S. This is done by considering the rank of the unfolded S viewed as a matrix A(S). The condition is: rank S ∈ {rank A(S), rank A(S) + 1}. In particular, srank S = rank S for S ∈ S d C n for the cases (d, n) ∈ {(3, 2), (4, 2), (3, 3)}. We discuss the analogs of the above results for border rank and best approximations of symmetric tensors.

… Colloquium on Computational Complexity ( …, 2005

The rigidity function of a matrix is defined as the minimum number of its entries that need to be changed in order to reduce the rank of the matrix to below a given parameter. Proving a strong enough lower bound on the rigidity of a matrix implies a nontrivial lower bound on the complexity of any linear circuit computing the set of linear forms associated with it. However, although it is shown that most matrices are rigid enough, no explicit construction of a rigid family of matrices is known.

arXiv (Cornell University), 2014

In this paper we introduce the notion of linear computability as a method of finding the Waring rank of forms. We use this notion to find infinitely many new examples which satisfy Strassen's Conjecture.

Linear Algebra and Its Applications, 2003

In representation theory, the problem of classifying pairs of matrices up to simultaneous similarity is used as a measure of complexity; classification problems containing it are called wild problems. We show in an explicit form that this problem contains all classification matrix problems given by quivers or posets. Then we prove that it does not contain (but is contained in) the problem of classifying threevalent tensors. Hence, all wild classification problems given by quivers or posets have the same complexity; moreover, a solution of any one of these problems implies a solution of each of the others. The problem of classifying three-valent tensors is more complicated.