As an emerging machine learning and information retrieval technique, the matrix completion has been successfully applied to solve many scientific applications, such as collaborative prediction in information retrieval, video completion in computer vision, etc. The matrix completion is to recover a low-rank matrix with a fraction of its entries arbitrarily corrupted. Instead of solving the popul...

We consider the problem of approximately reconstructing a partially-observed, approximately low-rank matrix. This problem has received much attention lately, mostly using the trace-norm as a surrogate to the rank. Here we study low-rank matrix reconstruction using both the trace-norm, as well as the less-studied max-norm, and present reconstruction guarantees based on existing analysis on the R...

This paper derives an inequality relating the p-norm of a positive 2×2 block matrix to the p-norm of the 2×2 matrix obtained by replacing each block by its p-norm. The inequality had been known for integer values of p, so the main contribution here is the extension to all values p ≥ 1. In a special case the result reproduces Hanner’s inequality. As an application in quantum information theory, ...

This paper derives an inequality relating the p-norm of a positive 2×2 block matrix to the p-norm of the 2×2 matrix obtained by replacing each block by its p-norm. The inequality had been known for integer values of p, so the main contribution here is the extension to all values p ≥ 1. In a special case the result reproduces Hanner’s inequality. A weaker inequality which applies also to non-pos...

We study the convergence of GMRES for linear algebraic systems with normal matrices. In particular, we explore the standard bound based on a min-max approximation problem on the discrete set of the matrix eigenvalues. This bound is sharp, i.e. it is attainable by the GMRES residual norm. The question is how to evaluate or estimate the standard bound, and if it is possible to characterize the GM...

The spectral p-norm of r-matrices generalizes the spectral 2-norm of 2-matrices. In 1911 Schur gave an upper bound on the spectral 2-norm of 2-matrices, which was extended in 1934 by Hardy, Littlewood, and Polya to r-matrices. Recently, Kolotilina, and independently the author, strengthened Schur’s bound for 2-matrices. The main result of this paper extends the latter result to r-matrices, ther...

The relationship between the stability margins of a 2-D digital filter (or discrete system) and the norm of the transition matrix of its minimum-norm realization is considered. Upper bounds on parameter variations, which guarantee the stability of a perturbed 2-D digital filter, are then derived in terms of the minimum norm. The results obtained are illustrated by two examples.

An iterative algorithm is constructed to solve the linear matrix equation pair AXB E, CXD F over generalized reflexive matrix X. When the matrix equation pair AXB E, CXD F is consistent over generalized reflexive matrix X, for any generalized reflexive initial iterative matrix X1, the generalized reflexive solution can be obtained by the iterative algorithm within finite iterative steps in the ...

The Schatten-p quasi-norm (0<p<1) is usually used to replace the standard nuclear norm in order to approximate the rank function more accurately. However, existing Schattenp quasi-norm minimization algorithms involve singular value decomposition (SVD) or eigenvalue decomposition (EVD) in each iteration, and thus may become very slow and impractical for large-scale problems. In this paper, we fi...

Let A be a complex m × n matrix. We find simple and good lower bounds for its spectral norm ‖A‖ = max{ ‖Ax‖ | x ∈ C, ‖x‖ = 1 } by choosing x smartly. Here ‖ · ‖ applied to a vector denotes the Euclidean norm.