Related papers: Nonnegative Rank vs. Binary Rank

It is easy to see that if $\cG$ is a non-abelian group of unitary matrices, then for no members $A$ and $B$ of $\cG$ can the rank of $AB-BA$ be one. We examine the consequences of the assumption that this rank is at most two for a general…

We consider a school choice matching model where the priorities for schools are represented by binary relations that may not be weak order. We focus on the (total order) extensions of the binary relations. We introduce a class of algorithms…

Binary classification involves predicting the label of an instance based on whether the model score for the positive class exceeds a threshold chosen based on the application requirements (e.g., maximizing recall for a precision bound).…

Paired comparison data, where users evaluate items in pairs, play a central role in ranking and preference learning tasks. While ordinal comparison data intuitively offer richer information than binary comparisons, this paper challenges…

Numerical analysis of data from international trade and ecological networks has shown that the non-linear fitness-complexity metric is the best candidate to rank nodes by importance in bipartite networks that exhibit a nested structure.…

A characteristic-dependent linear rank inequality is a linear inequality that holds by ranks of subspaces of a vector space over a finite field of determined characteristic, and does not in general hold over other characteristics. In this…

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…

We prove that any total boolean function of rank $r$ can be computed by a deterministic communication protocol of complexity $O(\sqrt{r} \cdot \log(r))$. Equivalently, any graph whose adjacency matrix has rank $r$ has chromatic number at…

We consider the set of $n\times n$ matrices with rational entries having numerator and denominator of size at most $H$ and obtain upper and lower bounds on the number of such matrices of a given rank and then apply them to count such…

In this paper, we propose and solve a low phase-rank approximation problem, which serves as a counterpart to the well-known low-rank approximation problem and the Schmidt-Mirsky theorem. More specifically, a nonzero complex number can be…

Nonlinear complexity, as an important measure for assessing the randomness of sequences, is defined as the length of the shortest feedback shift registers that can generate a given sequence. In this paper, the structure of n-periodic binary…

We study the noncommutative rank problem, ncRANK, of computing the rank of matrices with linear entries in $n$ noncommuting variables and the problem of noncommutative Rational Identity Testing, RIT, which is to decide if a given rational…

There has been continued interest in seeking a theorem describing optimal low-rank approximations to tensors of order 3 or higher, that parallels the Eckart-Young theorem for matrices. In this paper, we argue that the naive approach to this…

For the problems of low-rank matrix completion, the efficiency of the widely-used nuclear norm technique may be challenged under many circumstances, especially when certain basis coefficients are fixed, for example, the low-rank correlation…

This paper develops a general framework for conducting inference on the rank of an unknown matrix $\Pi_0$. A defining feature of our setup is the null hypothesis of the form $\mathrm H_0: \mathrm{rank}(\Pi_0)\le r$. The problem is of first…

We give an explicit upper bound for the number of equivalence classes of binary forms with rational integral coefficients of given degree and given discriminant, and with given splitting field. Further, we give an explicit upper bound for…

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.

Power nonnegative matrices are defined as complex matrices having at least one nonnegative integer power. We exploit the possibility of deriving a Perron Frobenius-like theory for these matrices, obtaining three main results and drawing…

Complex systems are high-dimensional nonlinear dynamical systems with intricate interactions among their constituents. To make interpretable predictions about their large-scale behavior, it is typically assumed, without a clear statement,…

While argument mining has achieved significant success in classifying argumentative relations between statements (support, attack, and neutral), we have a limited computational understanding of logical mechanisms that constitute those…