Maximum Number Of Nonzero Entries Research Articles

Tight Bounds for ℓ 1 Oblivious Subspace Embeddings

An \ell _p oblivious subspace embedding is a distribution over r \times n matrices \Pi such that for any fixed n \times d matrix A , \[ \Pr _{\Pi }[\textrm {for all }x, \ \Vert Ax\Vert _p \le \Vert \Pi Ax\Vert _p \le \kappa \Vert Ax\Vert _p] \ge 9/10, \] where r is the dimension of the embedding, \kappa is the distortion of the embedding, and for an n -dimensional vector y , \Vert y\Vert _p = (\sum _{i=1}^n |y_i|^p)^{1/p} is the \ell _p -norm. Another important property is the sparsity of \Pi , that is, the maximum number of non-zero entries per column, as this determines the running time of computing \Pi A . While for p = 2 there are nearly optimal tradeoffs in terms of the dimension, distortion, and sparsity, for the important case of 1 \le p \lt 2 , much less was known. In this article, we obtain nearly optimal tradeoffs for \ell _1 oblivious subspace embeddings, as well as new tradeoffs for 1 \lt p \lt 2 . Our main results are as follows: (1) We show for every 1 \le p \lt 2 , any oblivious subspace embedding with dimension r has distortion \[ \kappa = \Omega \left(\frac{1}{\left(\frac{1}{d}\right)^{1 / p} \log ^{2 / p}r + \left(\frac{r}{n}\right)^{1 / p - 1 / 2}}\right). \] When r = {\operatorname{poly}}(d) \ll n in applications, this gives a \kappa = \Omega (d^{1/p}\log ^{-2/p} d) lower bound, and shows the oblivious subspace embedding of Sohler and Woodruff (STOC, 2011) for p = 1 is optimal up to {\operatorname{poly}}(\log (d)) factors. (2) We give sparse oblivious subspace embeddings for every 1 \le p \lt 2 . Importantly, for p = 1 , we achieve r = O(d \log d) , \kappa = O(d \log d) and s = O(\log d) non-zero entries per column. The best previous construction with s \le {\operatorname{poly}}(\log d) is due to Woodruff and Zhang (COLT, 2013), giving \kappa = \Omega (d^2 {\operatorname{poly}}(\log d)) or \kappa = \Omega (d^{3/2} \sqrt {\log n} \cdot {\operatorname{poly}}(\log d)) and r \ge d \cdot {\operatorname{poly}}(\log d) ; in contrast our r = O(d \log d) and \kappa = O(d \log d) are optimal up to {\operatorname{poly}}(\log (d)) factors even for dense matrices. We also give (1) \ell _p oblivious subspace embeddings with an expected 1+\varepsilon number of non-zero entries per column for arbitrarily small \varepsilon \gt 0 , and (2) the first oblivious subspace embeddings for 1 \le p \lt 2 with O(1) -distortion and dimension independent of n . Oblivious subspace embeddings are crucial for distributed and streaming environments, as well as entrywise \ell _p low-rank approximation. Our results give improved algorithms for these applications.

Partial resampling to approximate covering integer programs†

We consider column‐sparse covering integer programs, a generalization of set cover. We develop a new rounding scheme based on the partial resampling variant of the Lovász Local Lemma developed by Harris and Srinivasan. This achieves an approximation ratio of , where amin is the minimum covering constraint and is the maximum ℓ1‐norm of any column of the covering matrix A (whose entries are scaled to lie in [0, 1]). With additional constraints on the variable sizes, we get an approximation ratio of (where is the maximum number of nonzero entries in any column of A). These results improve asymptotically over results of Srinivasan and results of Kolliopoulos and Young. We show nearly‐matching lower bounds. We also show that the rounding process leads to negative correlation among the variables.

0-1 matrices whose squares have bounded entries

Let Γ(n,k,t) be the set of 0-1 matrices of order n such that each entry of the k-th powers of these matrices is bounded by t. Let γ(n,k,t) be the maximum number of nonzero entries of a matrix in Γ(n,k,t). Given any positive integer t, we determine the values of γ(n,2,t) when n is sufficiently large. In addition, we discuss the structure of the matrices in Γ(n,2,t) attaining the maximum number of ones.

On k-idempotent 0-1 matrices

Let k≥2 be an integer. If a square 0-1 matrix A satisfies Ak=A, then A is said to be k-idempotent. In this paper, we give a characterization of k-idempotent 0-1 matrices. We also determine the maximum number of nonzero entries in k-idempotent 0-1 matrices of a given order as well as the k-idempotent 0-1 matrices attaining this maximum number.

0–1 matrices whose k-th powers have bounded entries

ABSTRACT Let be the set of 0–1 matrices of order n such that each entry of the k-th powers of these matrices is bounded by t. Let be the maximum number of nonzero entries of a matrix in . Given any positive integer t, we prove that for when n is sufficiently large, and this maximum number is attained at A if and only if A is permutation similar to the upper triangular tournament matrix of order n.

0-1 matrices with zero trace whose squares are 0-1 matrices

In this paper, we determine the maximum number of nonzero entries in 0-1 matrices of order n with zero trace whose squares are 0-1 matrices when n≥8. The extremal matrices attaining this maximum number are also characterized.

English

Let ℤ+ be the semiring of all nonnegative integers and A an m × n matrix over ℤ+. The rank of A is the smallest k such that A can be factored as an m × k matrix times a k×n matrix. The isolation number of A is the maximum number of nonzero entries in A such that no two are in any row or any column, and no two are in a 2 × 2 submatrix of all nonzero entries. We have that the isolation number of A is a lower bound of the rank of A. For A with isolation number k, we investigate the possible values of the rank of A and the Boolean rank of the support of A. So we obtain that the isolation number and the Boolean rank of the support of a given matrix are the same if and only if the isolation number is 1 or 2 only. We also determine a special type of m×n matrices whose isolation number is m. That is, those matrices are permutationally equivalent to a matrix A whose support contains a submatrix of a sum of the identity matrix and a tournament matrix.

Convertible Subspaces of Hessenberg-Type Matrices

We describe subspaces of generalized Hessenberg matrices where the determinant is convertible into the permanent by affixing ± signs. An explicit characterization of convertible Hessenberg-type matrices is presented. We conclude that convertible matrices with the maximum number of nonzero entries can be reduced to a basic set.

Extremal functions of forbidden multidimensional matrices

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 matrix A avoids anotherd-dimensional zero–one matrix P if no submatrix of A can be transformed to P by changing some ones to zeros. A fundamental problem is to study the maximum number of nonzero entries in a d-dimensional n×⋯×n matrix that avoids P. This maximum number, denoted by f(n,P,d), is called the extremal function.We advance the extremal theory of matrices in two directions. The methods that we use come from combinatorics, probability, and analysis. Firstly, we obtain non-trivial lower and upper bounds on f(n,P,d) when n is large for every d-dimensional block permutation matrix P. We establish the tight bound Θ(nd−1) on f(n,P,d) for every d-dimensional tuple permutation matrix P. This tight bound has the lowest possible order that an extremal function of a nontrivial matrix can ever achieve. Secondly, we show that the limit inferior of the sequence {f(n,P,d)nd−1} has a lower bound 2Ω(k1∕2) for a family of k×⋯×k permutation matrices P. We also improve the upper bound on the limit superior from 2O(klogk) to 2O(k) for all k×⋯×k permutation matrices and show that the new upper bound also holds for tuple permutation matrices.

Approximating Sparse Covering Integer Programs Online

A covering integer program (CIP) is a mathematical program of the form min{c⊤ x ∣ Ax ≥ 1, 0 ≤ x ≤ u, x ∈ ℤn}, where all entries in A, c, u are nonnegative. In the online setting, the constraints (i.e., the rows of the constraint matrix A) arrive over time, and the algorithm can only increase the coordinates of x to maintain feasibility. As an intermediate step, we consider solving the covering linear program (CLP) online, where the integrality constraints are dropped. Our main results are (a) an O(log k)-competitive online algorithm for solving the CLP, and (b) an O(log k · log l)-competitive randomized online algorithm for solving the CIP. Here k ≤ n and l ≤ m respectively denote the maximum number of nonzero entries in any row and column of the constraint matrix A. Our algorithm is based on the online primal-dual paradigm, where a novel ingredient is to allow dual variables to increase and decrease throughout the course of the algorithm. It is known that this result is the best possible for polynomial-time online algorithms, even in the special case of set cover (where all entries in A, c, and u are 0 or 1.

Integrally normalizable matrices and zero–nonzero patterns

The problem of determining whether or not an n×n integer matrix is diagonally similar to an integer matrix with n−1 off-diagonal entries equal to 1 is studied. Such a matrix is called integrally normalizable, and a zero–nonzero pattern is integrally normalizable if each matrix with this zero–nonzero pattern is integrally normalizable with respect to the same set of n−1 off-diagonal entries. Matrices that are integrally normalizable with respect to a fixed spanning tree, and integrally normalizable zero–nonzero patterns are characterized. The maximum number of nonzero entries in an n×n integrally normalizable zero–nonzero pattern is shown to be n2+3n−22. Extensions to matrices over other integral domains are also presented.

Nilpotent sign patterns of a given index

A sign pattern is said to be nilpotent of index k if all real matrices in its qualitative class are nilpotent and their maximum nilpotent index equals k. In this paper, we characterize sign patterns that are nilpotent of a given index k. The maximum number of nonzero entries in such sign patterns of a given order is determined as well as the sign patterns with this maximum number of nonzero entries.

Iterative oblique projection onto convex sets and the split feasibility problem

Let C andQ be nonempty closedconvex sets in RNand RM,respectively, and Aan Mby Nreal matrix. The split feasibility problem (SFP) is to find x ∊ C withAx ∊ Q,if such xexist. An iterative method for solving the SFP, called the CQ algorithm,has the following iterative step: xk+1 = P C (xk + γAT (P Q − I)Axk), where γ ∊ (0, 2∖L) withL the largest eigenvalueof the matrix ATAand PC andPQ denote the orthogonalprojections onto C andQ, respectively; thatis, PCx minimizes||c − x||,over all c ∊ C.The CQalgorithm converges to a solution of the SFP, or, more generally, to a minimizer of ||PQAc − Ac||over c inC, whenever such exist.The CQalgorithm involves only the orthogonal projections ontoC andQ,which we shall assume are easily calculated, and involves no matrix inverses. IfAis normalized so that each row has length one, thenL doesnot exceed the maximum number of nonzero entries in any column ofA, which provides ahelpful estimate of Lfor sparse matrices.Particular cases of the CQalgorithm are the Landweber and projected Landweber methods forobtaining exact or approximate solutions of the linear equationsAx = b;the algebraic reconstruction technique of Gordon, Bender and Hermanis a particular case of a block-iterative version of the CQalgorithm.One application of the CQ algorithm that is the subject of ongoingwork is dynamic emission tomographic image reconstruction, in whichthe vectorx is theconcatenation of several images corresponding to successive discrete times. The matrixA andthe set Qcan then be selected to impose constraints on the behaviour over time of theintensities at fixed voxels, as well as to require consistency (or near consistency)with measured data.