This paper is concerned with iterative solution methods for large linear systems of equations with a matrix of ill-determined rank and an error-contaminated right-hand side. The numerical solution is delicate, because the matrix is very ill-conditioned and may be singular. It is natural to require that the computed iterates live in the range of the matrix when the latter is symmetric, because t...

The low-rank matrix completion problem is a fundamental machine learning and data mining problem with many important applications. The standard low-rank matrix completion methods relax the rank minimization problem by the trace norm minimization. However, this relaxation may make the solution seriously deviate from the original solution. Meanwhile, most completion methods minimize the squared p...

This paper proposes a robust reduced-rank scheme for adaptive beamforming based on joint iterative optimization (JIO) of adaptive filters. The scheme provides an efficient way to deal with filters with large number of elements. It consists of a bank of full-rank adaptive filters that forms a transformation matrix and an adaptive reduced-rank filter that operates at the output of the bank of fil...

this paper addresses adaptive observer design problem for joint estimation of the states and unknown parameters for a class of nonlinear systems which satisfying one-sided lipschitz and quadratic inner bounded conditions. it’s shown that the stability of the proposed observer is related to finding solutions to a quadratic inequality consists of state and parameter errors. a coordinate transform...

Matrix completion, the problem of completing missing entries in a data matrix with low-dimensional structure (such as rank), has seen many fruitful approaches and analyses. Tensor completion is tensor analog that attempts to impute from similar low-rank type assumptions. In this paper, we study when sampling pattern deterministic possibly non-uniform. We first propose an efficient weighted High...

Abstract: In this paper we resume some results concerned our work about least-squares approximation on GaussLobatto points. We present explicit formulas for discrete orthogonal polynomials and give the three-term recurrence relation to construct such polynomials. We also show that the normal matrix on this set of nodes can be factorized as the sum of two symmetric matrices: a full rank matrix w...

Intuitively, if a density operator has small rank, then it should be easier to estimate from experimental data, since in this case only a few eigenvectors need to be learned. We prove two complementary results that confirm this intuition. Firstly, we show that a low-rank density matrix can be estimated using fewer copies of the state, i.e. the sample complexity of tomography decreases with the ...

We consider the following problem: given two matrices A and B, find a rank-r approximation of their product ATB. This type of linear algebra problem has many applications in the machine learning and statistics domain. For example, if A = B, then this general problem reduces to the well-known problem of finding principal components of a given data matrix. Another example is the low-rank approxim...

It is a long open problem to combinatorially characterize the 3D bar-joint rigidity of graphs. The problem is at the intersection of combinatorics and algebraic geometry, and crops up in practical algorithmic applications ranging from mechanical computer aided design to molecular modeling. The problem is equivalent to combinatorially determining the generic rank of the 3D bar-joint rigidity mat...