Related papers: Sign patterns with minimum rank 3 and point-line c…
We introduce a novel concept of rank for subsets of finite metric spaces E^n_q (the set of all n-dimensional vectors over an alphabet of size q) equipped with the Hamming distance, where the rank R(A) of a subset A is defined as the number…
The low-rank matrix optimization with affine set (rank-MOA) is to minimize a continuously differentiable function over a low-rank set intersecting with an affine set. Under some suitable assumptions, the intersection rule of the Fr\'{e}chet…
An interval matrix is a matrix whose entries are intervals in the set of real numbers. We generalize this concept, which has been broadly studied, to other fields. Precisely we define a rational interval matrix to be a matrix whose entries…
The nonnegative rank of an entrywise nonnegative matrix A of size mxn is the smallest integer r such that A can be written as A=UV where U is mxr and V is rxn and U and V are both nonnegative. The nonnegative rank arises in different areas…
The work considers an equivalence relation in the set of all $n\times m$ matrices with entries in the set $[p]=\{ 0,1,\ldots , p-1 \}$. In each element of the factor-set generated by this relation, we define the concept of canonical matrix,…
A sign pattern is a matrix whose entries are from the set $\{+,-, 0\}$. A square sign pattern $A$ is called sign $k$-potent if $k$ is the smallest positive integer for which $A^{k+1}=A$, and for $k=1$, $A$ is called sign idempotent. In…
The notion of fractional minimal rank of a partial matrix is introduced, a quantity that lies between the triangular minimal rank and the minimal rank of a partial matrix. The fractional minimal rank of partial matrices whose bipartite…
The minimum rank of a simple graph $G$ is defined to be the smallest possible rank over all symmetric real matrices whose $ij$th entry (for $i\neq j$) is nonzero whenever $\{i,j\}$ is an edge in $G$ and is zero otherwise. Minimum rank is a…
The signature of a path is a sequence of tensors which allows to uniquely reconstruct the path. In this paper we propose a systematic study of basic properties of signature tensors, starting from their rank, symmetries and conciseness. We…
We prove new upper and lower bounds on $\epsilon$-approximate sign-rank, a relaxation of sign-rank introduced by Chornomaz, Moran, and Waknine (STOC 2025). We show that every $m \times n$ sign matrix with approximate sign-rank $d$ contains…
We prove a characterization of the structural conditions on matrices of sign-rank 3 and unit disk graphs (UDGs) which permit constant-cost public-coin randomized communication protocols. Therefore, under these conditions, these graphs also…
Low-rank matrix completion addresses the problem of completing a matrix from a certain set of generic specified entries. Over the complex numbers a matrix with a given entry pattern can be uniquely completed to a specific rank, called the…
Let $F$ be a $k\times \ell$ (0,1)-matrix. A matrix is simple if it is a (0,1)-matrix with no repeated columns. A (0,1)-matrix $A$ is said to have a $F$ as a configuration if there is a submatrix of $A$ which is a row and column permutation…
The real minimum skew rank of a simple graph G is the smallest possible rank among all real skew symmetric matrices, whose (i,j)-entry (for i not equal to j) is nonzero whenever {i, j} is an edge in G and is zero otherwise. In this paper we…
Matrices of (approximate) low rank are pervasive in data science, appearing in recommender systems, movie preferences, topic models, medical records, and genomics. While there is a vast literature on how to exploit low rank structure in…
The rank of a graph is defined to be the rank of its adjacency matrix. A graph is called reduced if it has no isolated vertices and no two vertices with the same set of neighbors. Akbari, Cameron, and Khosrovshahi conjectured that the…
Let $R$ and $S$ be two sequences of nonnegative integers in nonincreasing order and with the same sum, and let ${\cal A}(R,S)$ be the class of all $(0,1)$-matrices having row sum $R$ and column sum $S$. For a positive integer $t$, the…
In this lecture note, we discuss a fundamental concept, referred to as the {\it characteristic rank}, which suggests a general framework for characterizing the basic properties of various low-dimensional models used in signal processing.…
Generalized inverses are important in statistics and other areas of applied matrix algebra. A \emph{generalized inverse} of a real matrix $A$ is a matrix $H$ that satisfies the Moore-Penrose (M-P) property $AHA=A$. If $H$ also satisfies the…
We study modeling and identification of processes with a spectral density matrix of low rank. Equivalently, we consider processes having an innovation of reduced dimension for which Prediction Error Methods (PEM) algorithms are not directly…