Related papers: An exact characterization of saturation for permut…
We propose layer saturation - a simple, online-computable method for analyzing the information processing in neural networks. First, we show that a layer's output can be restricted to the eigenspace of its variance matrix without…
This is the first of two papers to describe a matrix sparsification algorithm that takes a general real or complex matrix as input and produces a sparse output matrix of the same size. The non-zero entries in the output are chosen to…
The 0-1 matrix A contains a 0-1 matrix M if some submatrix of A can be transformed into M by changing some ones to zeroes. If A does not contain M, then A avoids M. Let ex(n,M) be the maximum number of ones in an n x n 0-1 matrix that…
Rearranging the rows or columns of a sparse matrix using an appropriate ordering can significantly reduce fill-ins, i.e., new nonzeros introduced during matrix factorization, decreasing memory usage and runtime. However, finding an ordering…
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…
We show that, for a positive integer $r$, every minimal 1-saturating set in ${\rm PG}(r-1,2)$ of size at least ${11/36} 2^r+3$ is either a complete cap or can be obtained from a complete cap $S$ by fixing some $s\in S$ and replacing every…
Consider a finite sequence of permutations of the elements 1,...,n, with the property that each element changes its position by at most 1 from any permutation to the next. We call such a sequence a tangle, and we define a move of element i…
Denote by $A_n$ the set of square $(0,1)$ matrices of order $n$. The set $A_n$, $n\le8$, is partitioned into row/column permutation equivalence classes enabling derivation of various facts by simple counting. For example, the number of…
Machines whose main purpose is to permute and sort data are studied. The sets of permutations that can arise are analysed by means of finite automata and avoided pattern techniques. Conditions are given for these sets being enumerated by…
A superpermutation is a sequence that contains every permutation of $n$ distinct symbols as a contiguous substring. For instance, a valid example for three symbols is a sequence that contains all six permutations. This paper introduces a…
Matrix completion is a modern missing data problem where both the missing structure and the underlying parameter are high dimensional. Although missing structure is a key component to any missing data problems, existing matrix completion…
A family $\cF \subseteq 2^{[n]}$ saturates the monotone decreasing property $\cP$ if $\cF$ satisfies $\cP$ and one cannot add any set to $\cF$ such that property $\cP$ is still satisfied by the resulting family. We address the problem of…
For a fixed poset $\mathcal P$ we say that a family $\mathcal F\subseteq\mathcal P([n])$ is $\mathcal P$-saturated if it does not contain an induced copy of $\mathcal P$, but whenever we add a new set to $\mathcal F$, we form an induced…
A sign pattern matrix is a matrix whose entries are from the set $\{+,-,0\}$. If $A$ is an $m\times n$ sign pattern matrix, the qualitative class of $A$, denoted $Q(A)$, is the set of all real $m\times n$ matrices $B=[b_{i,j}]$ with…
Pattern avoidance is a central topic in graph theory and combinatorics. Pattern avoidance in matrices has applications in computer science and engineering, such as robot motion planning and VLSI circuit design. A $d$-dimensional zero-one…
Zeros of rational transfer function matrices $R(\lambda)$ are the eigenvalues of associated polynomial system matrices $P(\lambda)$, under minimality conditions. In this paper we define a structured condition number for a simple eigenvalue…
Matrix completion is a classical problem in data science wherein one attempts to reconstruct a low-rank matrix while only observing some subset of the entries. Previous authors have phrased this problem as a nuclear norm minimization…
For a given positive integer $k$ we say that a family of subsets of $[n]$ is $k$-antichain saturated if it does not contain $k$ pairwise incomparable sets, but whenever we add to it a new set, we do find $k$ such sets. The size of the…
A $d$-dimensional zero-one matrix $A$ avoids another $d$-dimensional zero-one matrix $P$ if no submatrix of $A$ can be transformed to $P$ by changing some ones to zeroes. Let $f(n,P,d)$ denote the maximum number of ones in a $d$-dimensional…
Packing density is a permutation occurrence statistic which describes the maximal number of permutations of a given type that can occur in another permutation. In this article we focus on containment of sets of permutations. Although this…