We focus the use of row sampling for approximating matrix algorithms. We give applications to matrix multipication; sparse matrix reconstruction; and, l2 regression. For a matrix A ∈ R m×d which represents m points in d ≪ m dimensions, all of these tasks can be achieved in O(md) via the singular value decomposition (SVD). For appropriate row-sampling probabilities (which typically depend on the...

Necessary and sufficient conditions are given for a substochastic semigroup on L obtained through the Kato–Voigt perturbation theorem to be either stochastic or strongly stable. We show how such semigroups are related to piecewise deterministic Markov process, provide a probabilistic interpretation of our results, and apply them to fragmentation equations.

A binary matrix has the Consecutive Ones Property (C1P) if there exists a permutation of its columns (i.e. a sequence of column swappings) such that in the resulting matrix the 1s are consecutive in every row. A Minimal Conflicting Set (MCS) of rows is a set of rows R that does not have the C1P, but such that any proper subset of R has the C1P. In [5], Chauve et al. gave a O(∆m(n+m+ e)) time al...

We describe a new algorithm called Frequent Directions for deterministic matrix sketching in the row-updates model. The algorithm is presented an arbitrary input matrix A ∈ Rn×d one row at a time. It performed O(d`) operations per row and maintains a sketch matrix B ∈ R`×d such that for any k < ` ‖AA−BB‖2 ≤ ‖A−Ak‖F /(`− k) and ‖A− πBk(A)‖F ≤ ( 1 + k `− k ) ‖A−Ak‖F . Here, Ak stands for the mini...

In this paper, a numerical method for nding minimal solution of a mn fullyfuzzy linear system of the form Ax = b based on pseudo inverse calculation,is given when the central matrix of coecients is row full rank or column fullrank, and where A~ is a non-negative fuzzy mn matrix, the unknown vectorx is a vector consisting of n non-negative fuzzy numbers and the constant b isa vector consisting o...

The eccentricity matrix of a connected graph $G$, denoted by $\mathcal{E}(G)$, is obtained from the distance $G$ keeping largest nonzero entries in each row and column leaving zeros remaining ones. $\mathcal{E}$-eigenvalues are eigenvalues $\mathcal{E}(G)$. modulus an eigenvalue $\mathcal{E}$-spectral radius $G$. $\mathcal{E}$-energy sum absolute values all In this article, we study some extrem...

An assignment problem is the optimization problem of finding, in an m by n matrix of nonnegative real numbers, k entries, no two in the same row or column, such that their sum is minimal. Such an optimization problem is called a random assignment problem if the matrix entries are random variables. We give a formula for the expected value of the optimal k-assignment in a matrix where some of the...

In this note I give a proof of the existence of a Smith normal form for matrices with integer entries. The existence of a good basis for a lattice with a finite index sublattice is a consequence of the Smith normal form. I conclude with an easy application to toric varieties. Theorem 1. Let A be a matrix with integer entries. Then there exist integer-values invertible matrices C and B such that...

where Mk,n is the set of all k-by-n matrices with coefficients in {0, 1} and the variables y l are interpreted as follows: y i l = 1 if and only if z i = l. The matrix (y j) can be seen as a {0, 1} assignment matrix where each column contains exactly one coefficient equal to 1 while h denotes the index of the lowest row that is not identically equal to the zero row (cf. Fig. 1). Another variant...