In this paper, we show that every separable simple tracially approximately divisible C ? $C^*$ -algebra has strict comparison, and it is either purely infinite or stable rank one. As a consequence, (non-unital) finite Z ${\cal Z}$ -stable

We consider the problem of recovering low-rank matrices from random rank-one measurements, which spans numerous applications including covariance sketching, phase retrieval, quantum state tomography, and learning shallow polynomial neural networks, among others. Our approach is to directly estimate factor by minimizing a nonconvex least-squares loss function via vanilla gradient descent, follow...

This paper gives normwise and componentwise perturbation analyses for the Q-factor of the QR factorization of the matrix A with full column rank when A suffers from an additive perturbation. Rigorous perturbation bounds are derived on the projections of the perturbation of the Q-factor in the range of A and its orthogonal complement. These bounds overcome a serious shortcoming of the first-orde...

In this paper, we propose an efficient and scalable low rank matrix completion algorithm. The key idea is to extend orthogonal matching pursuit method from the vector case to the matrix case. We further propose an economic version of our algorithm by introducing a novel weight updating rule to reduce the time and storage complexity. Both versions are computationally inexpensive for each matrix ...

We study the recovery of Hermitian low rank matrices X ∈ Cn×n from undersampled measurements via nuclear norm minimization. We consider the particular scenario where the measurements are Frobenius inner products with random rank-one matrices of the form ajaj for some measurement vectors a1, . . . , am, i.e., the measurements are given by yj = tr(Xaja ∗ j ). The case where the matrix X = xx ∗ to...

in this paper, elzaki homotopy perturbation method is employed for solving linear and nonlinear differential equations with a variable coffecient. this method is a combination of elzaki transform and homotopy perturbation method. the aim of using elzaki transform is to overcome the deficiencies that mainly caused by unsatised conditions in some semi-analytical methods such as homotopy perturbat...