Related papers: The Complexity of Tensor Rank
We show that completeness at higher levels of the theory of the reals is a robust notion (under changing the signature and bounding the domain of the quantifiers). This mends recognized gaps in the hierarchy, and leads to stronger…
We develop a framework to analyse invariant decompositions of elements of tensor product spaces. Namely, we define an invariant decomposition with indices arranged on a simplicial complex, and which is explicitly invariant under a group…
This paper presents a combinatorial analog of topological complexity for finite spaces. We demonstrate that this coincides with the genuine topological complexity of the original finite space, and constitutes an upper bound for the…
We show that computing the minimum rank of a sign pattern matrix is NP hard. Our proof is based on a simple but useful connection between minimum ranks of sign pattern matrices and the stretchability problem for pseudolines arrangements. In…
Complexity theory provides a wealth of complexity classes for analyzing the complexity of decision and counting problems. Despite the practical relevance of enumeration problems, the tools provided by complexity theory for this important…
The relationship between the complexity classes P and NP is a question that has not yet been answered by the Theory of Computation. The existence of a language in NP, proven not to belong to P, is sufficient evidence to establish the…
This survey aims at demonstrating that the structure of precedence constraints plays a tremendous role on the complexity of scheduling problems. Indeed many problems can be NP-hard when considering general precedence constraints, while they…
Sequential data such as time series, video, or text can be challenging to analyse as the ordered structure gives rise to complex dependencies. At the heart of this is non-commutativity, in the sense that reordering the elements of a…
We investigate the problem of deciding whether a system of linear equations, together with divisibility conditions on the variables, has a solution over holomorphy subrings of global fields. We obtain decidability results when we allow…
Recent studies showed that hardness, a complex property, can be calculated using very simple approaches or even analytical formulae. These form the basis for evaluating controversial experimental results (as we illustrate for…
In translation surfaces of finite area (corresponding to holomorphic differentials), directions of saddle connections are dense in the unit circle. On the contrary, saddle connections are fewer in translation surfaces with poles…
Let S = <s_1, s_2, s_3, ..., s_n> be a given vector of n real numbers. The rank of a real z with respect to S is defined as the number of elements s_i in S such that s_i is less than or equal to z. We consider the following decision…
We introduce a first-order theory of finite full binary trees and then identify decidable and undecidable fragments of this theory. We show that the analogue of Hilbert`s 10th Problem is undecidable by constructing a many-to-one reduction…
In this article the notion of the nondecreasing (ND) rank of a matrix or tensor is introduced. A tensor has an ND rank of r if it can be represented as a sum of r outer products of vectors, with each vector satisfying a monotonicity…
We establish new upper bounds about symmetric bilinear complexity in any extension of finite fields. Note that these bounds are not asymptotical but uniform. Moreover we give examples of Shimura curves that do not descend over their field…
It is shown that determining whether a quantum computation has a non-zero probability of accepting is at least as hard as the polynomial time hierarchy. This hardness result also applies to determining in general whether a given quantum…
This paper addresses the detection of a low rank high-dimensional tensor corrupted by an additive complex Gaussian noise. In the asymptotic regime where all the dimensions of the tensor converge towards $+\infty$ at the same rate, existing…
A novel lower bound is introduced for the full rank probability of random finite field matrices, where a number of elements with known location are identically zero, and remaining elements are chosen independently of each other, uniformly…
A theory of gravity with torsion is examined in which the torsion tensor is constructed from the exterior derivative of an antisymmetric rank two potential plus the dual of the gradient of a scalar field. Field equations for the theory are…
The notion of a tensor captures three great ideas: equivariance, multilinearity, separability. But trying to be three things at once makes the notion difficult to understand. We will explain tensors in an accessible and elementary way…