In this paper we propose an explicit solution to the polynomial least squares approximation problem on Chebyshev extrema nodes. We also show that the inverse of the normal matrix on this set of nodes can be represented as the sum of two symmetric matrices: a full rank matrix which admits a Cholesky factorization and a 2-rank matrix. Finally we discuss the numerical properties of the proposed fo...

This work is concerned with the numerical solution of large-scale linear matrix equations A1XB T 1 + · · ·+ AKXB K = C. The most straightforward approach computes X ∈ Rm×n from the solution of an mn×mn linear system, typically limiting the feasible values of m,n to a few hundreds at most. Our new approach exploits the fact that X can often be well approximated by a low-rank matrix. It combines ...

This work is concerned with the numerical solution of large-scale linear matrix equations A1XB T 1 + · · ·+ AKXB K = C. The most straightforward approach computes X ∈ Rm×n from the solution of an mn×mn linear system, typically limiting the feasible values of m,n to a few hundreds at most. Our new approach exploits the fact that X can often be well approximated by a low-rank matrix. It combines ...

In this paper, the important problem of single-channel noise reduction is treated from a new perspective. The problem is posed as a filtering problem based on joint diagonalization of the covariance matrices of the desired and noise signals. More specifically, the eigenvectors from the joint diagonalization corresponding to the least significant eigenvalues are used to form a filter, which effe...

For a positive integer k, the rank-k numerical range Λk(A) of an operator A acting on a Hilbert space H of dimension at least k is the set of scalars λ such that PAP = λP for some rank k orthogonal projection P . In this paper, a close connection between low rank perturbation of an operator A and Λk(A) is established. In particular, for 1 ≤ r < k it is shown that Λk(A) ⊆ Λk−r(A + F ) for any op...

A random matrix is likely to be well conditioned, and motivated by this well known property we employ random matrix multipliers to advance some fundamental matrix computations. This includes numerical stabilization of Gaussian elimination with no pivoting as well as block Gaussian elimination, approximation of the leading and trailing singular spaces of an ill conditioned matrix, associated wit...

A random matrix is likely to be well conditioned, and motivated by this well known property we employ random matrix multipliers to advance some fundamental matrix computations. This includes numerical stabilization of Gaussian elimination with no pivoting as well as block Gaussian elimination, approximation of the leading and trailing singular spaces of an ill conditioned matrix, associated wit...

A random matrix is likely to be well conditioned, and motivated by this well known property we employ random matrix multipliers to advance some fundamental matrix computations. This includes numerical stabilization of Gaussian elimination with no pivoting as well as block Gaussian elimination, approximation of the leading and trailing singular spaces of an ill conditioned matrix, associated wit...

We present a unified framework for low-rank matrix estimation with nonconvex penalty. A proximal gradient homotopy algorithm is developed to solve the proposed optimization problem. Theoretically, we first prove that the proposed estimator attains a faster statistical rate than the traditional low-rank matrix estimator with nuclear norm penalty. Moreover, we rigorously show that under a certain...

Matrix rank minimization problem is widely applicable in many fields such as control, signal processing and system identification. However, the problem is in general NP-hard, and it is computationally hard to solve directly in practice. In this paper, we provide a new kind of approximation functions for the rank of matrix, and the corresponding approximation problems can be used to approximate ...