Related papers: The Complexity of Tensor Rank
We propose a numerical method to obtain an adequate value for the upper bound on the rank for the tensor completion problem on the variety of third-order tensors of bounded tensor-train rank. The method is inspired by the parametrization of…
In the bibliography a certain confusion arises in what regards to the classification of the gravitational theories into scalar-tensor theories and general relativity with a scalar field either minimally or non-minimally coupled to matter.…
We provide new upper and lower bounds on the minimum possible ratio of the spectral and Frobenius norms of a (partially) symmetric tensor. In the particular case of general tensors our result recovers a known upper bound. For symmetric…
A tensor is a multi-way array that can represent, in addition to a data set, the expression of a joint law or a multivariate function. As such it contains the description of the interactions between the variables corresponding to each of…
Low-rank tensor approximations have shown great potential for uncertainty quantification in high dimensions, for example, to build surrogate models that can be used to speed up large-scale inference problems (Eigel et al., Inverse Problems…
An attempt of a new kind of complexity anthropology is considered.
The log-rank conjecture is one of the fundamental open problems in communication complexity. It speculates that the deterministic communication complexity of any two-party function is equal to the log of the rank of its associated matrix,…
Matrices of rank at most k are defined by the vanishing of polynomials of degree k + 1 in their entries (namely, their (k + 1)-times-(k + 1)-subdeterminants), regardless of the size of the matrix. We prove a qualitative analogue of this…
We study the computational complexity of an important property of simple, regular and weighted games, which is decisiveness. We show that this concept can naturally be represented in the context of hypergraph theory, and that decisiveness…
Let $T$ be a real tensor of (real) rank $r$. $T$ is 'identifiable' when it has a unique decomposition in terms of rank $1$ tensors. There are cases in which the identifiability fails over the complex field, for general tensors of rank $r$.…
The rank of a tensor is analyzed in context of quantum entanglement. A pure quantum state $\bf v$ of a composite system consisting of $d$ subsystems with $n$ levels each is viewed as a vector in the $d$-fold tensor product of…
We show that determining the crossing number of a link is NP-hard. For some weaker notions of link equivalence, we also show NP-completeness.
We determine the rank of a random matrix over an arbitrary field with prescribed numbers of non-zero entries in each row and column. As an application we obtain a formula for the rate of low-density parity check codes. This formula…
Asymptotic tensor rank is notoriously difficult to determine. Indeed, determining its value for the $2\times 2$ matrix multiplication tensor would determine the matrix multiplication exponent, a long-standing open problem. On the other…
We give a short proof for a well-known formula for the rank of a $G$-crossed braided extension of a modular tensor category.
In this note, we leverage some of our results from arXiv:1706.06319 to produce a concise and rigorous proof for the complexity of the generalized MinRank Problem in the under-defined and well-defined case. Our main theorem recovers and…
We describe in the space of binary forms of degree d the strata of forms having constant rank. We also give a simple algorithm to determine the rank of a given form.
There are close relations between tripartite tensors with bounded geometric ranks and linear determinantal varieties with bounded codimensions. We study linear determinantal varieties with bounded codimensions, and prove upper bounds of the…
Kruskal's theorem states that a sum of product tensors constitutes a unique tensor rank decomposition if the so-called k-ranks of the product tensors are large. In this work, we propose a conjecture in which the k-rank condition of…
We study the decomposability and the subdifferential of the tensor nuclear norm. Both concepts are well understood and widely applied in matrices but remain unclear for higher-order tensors. We show that the tensor nuclear norm admits a…