The modified Gram–Schmidt algorithm is a well–known and widely used procedure to orthogonalize the column vectors of a given matrix. When applied to ill–conditioned matrices in floating point arithmetic, the orthogonality among the computed vectors may be lost. In this work, we propose an a posteriori reorthogonalization technique based on a rank–k update of the computed vectors. The level of o...

The modified Gram–Schmidt algorithm is a well-known and widely used procedure to orthogonalize the column vectors of a given matrix. When applied to ill-conditioned matrices in floating point arithmetic, the orthogonality among the computed vectors may be lost. In this work, we propose an a posteriori reorthogonalization technique based on a rank-k update of the computed vectors. The level of o...

Low-rank factorization is used in many areas of computer science where one performs spectral analysis on large sensitive data stored in the form of matrices. In this paper, we study differentially private low-rank factorization of a matrix with respect to the spectral norm in the turnstile update model. In this problem, given an input matrix A ∈ Rm×n updated in the turnstile manner and a target...

In this paper, we provide local and global convergence guarantees for recovering CP (Candecomp/Parafac) tensor decomposition. The main step of the proposed algorithm is a simple alternating rank-1 update which is the alternating version of the tensor power iteration adapted for asymmetric tensors. Local convergence guarantees are established for third order tensors of rank k in d dimensions, wh...

the rank-k numerical range has a close connection to the construction of quantum error correction code for a noisy quantum channel. for noisy quantum channel, a quantum error correcting code of dimension k exists if and only if the associated joint rank-k numerical range is non-empty. in this paper the notion of joint rank-k numerical range is generalized and some statements of [2011, generaliz...

3 Adapting the Covariance Matrix 10 3.1 Estimating the Covariance Matrix From Scratch . . . . . . . . . . . . . . . . 10 3.2 Rank-μ-Update . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 3.3 Rank-One-Update . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 3.3.1 A Different Viewpoint . . . . . . . . . . . . . . . . . . . . . . . . . 14 3.3.2 Cumulation: Uti...

The rank-k numerical range has a close connection to the construction of quantum error correction code for a noisy quantum channel. For noisy quantum channel, a quantum error correcting code of dimension k exists if and only if the associated joint rank-k numerical range is non-empty. In this paper the notion of joint rank-k numerical range is generalized and some statements of [2011, Generaliz...

The ABS methods, introduced by Abaffy, Broyden and Spedicato, are direct iteration methods for solving a linear system where the ith iterate satisfies the first i equations, therefore a system of m equations is solved in at most m steps. Recently, we have presented a new approach to devise a class of ABS-type methods for solving full row rank systems [K. Amini, N. Mahdavi-Amiri, M. R. Peyghami,...