Zero-sum games are two player games played on a matrix M ∈ Matm×n(R). The row player, denoted R, chooses a row i ∈ [m] and the column player C chooses a column j ∈ [n], simultaneously. The payoff to the row player is Mij and the payoff to the column player is −Mij (hence the game is “zero-sum”). We can also consider randomized strategies, where R chooses a probability distribution p on [m], whi...

Let $mathbf{c}_0$ be the real vector space of all real sequences which converge to zero. For every $x,yin mathbf{c}_0$, it is said that $y$ is block diagonal majorized by $x$ (written $yprec_b x$) if there exists a block diagonal row stochastic matrix $R$ such that $y=Rx$. In this paper we find the possible structure of linear functions $T:mathbf{c}_0rightarrow mathbf{c}_0$ preserving $prec_b$.

A zero sum game is a simultaneous move game between 2 players. Such a game is represented by a matrix A ∈ Rm×n. The strategies of the “row” (resp. “column”) player are the rows (resp. columns) of A. If the row player plays strategy i ∈ [m] and the column player plays j ∈ [n] then the outcome is Aij . 1 Interpret this as that the row player pays Aij amount of money to the column player, therefor...

Many optimization algorithms involve repeated processing of a fixed set of linear constraints. If we pre-process the constraint matrix A to make it sparser, algebraic operations should become faster. In many applications there is a priori information about the likelihood that each column will appear in a basis, which can be expressed as weights on the columns. This leads to considering the Weig...

We consider the problem of sparse matrix multiplication by the column row method in a distributed setting where the matrix product is not necessarily sparse. We present a surprisingly simple method for “consistent” parallel processing of sparse outer products (column-row vector products) over several processors, in a communication-avoiding setting where each processor has a copy of the input. T...

Matrix extension with symmetry is to find a unitary square matrix P of 2π-periodic trigonometric polynomials with symmetry such that the first row of P is a given row vector p of 2πperiodic trigonometric polynomials with symmetry satisfying pp = 1. Matrix extension plays a fundamental role in many areas such electronic engineering, system sciences, wavelet analysis, and applied mathematics. In ...

The sparse matrix-vector multiplication (SpMxV) is a kernel operation widely used in iterative linear solvers. The same sparse matrix is multiplied by a dense vector repeatedly in these solvers. Matrices with irregular sparsity patterns make it difficult to utilize cache locality effectively in SpMxV computations. In this work, we investigate singleand multiple-SpMxV frameworks for exploiting c...

This note considers a problem of minimum length scheduling for a set of messages subject to precedence constraints for switching and communication networks. The problem was first studied by Barcaccia, Bonuccelli, and Di Iannii [1]. We consider a network with n inputs and n outputs. The messages to be sent are represented by an n × n matrix D = [di j], the traffic matrix, whose entries are nonne...

Given n samples X1,X2, . . . ,Xn from N(0, ), we are interested in estimating the p×p precision matrix = −1; we assume is sparse in that each row has relatively few nonzeros. We propose Partial Correlation Screening (PCS) as a new row-by-row approach. To estimate the ith row of , 1 ≤ i ≤ p, PCS uses a Screen step and a Clean step. In the Screen step, PCS recruits a (small) subset of indices usi...

The sparse matrix-vector multiplication (SpMxV) is a kernel operation widely used in iterative linear solvers. The same sparse matrix is multiplied by a dense vector repeatedly in these solvers. Matrices with irregular sparsity patterns make it difficult to utilize cache locality effectively in SpMxV computations. In this work, we investigate singleand multiple-SpMxV frameworks for exploiting c...