New perturbation theorems are proved for simultaneous bases of singular subspaces of real matrices. These results improve the absolute bounds previously obtained in [6] for general (complex) matrices. Unlike previous results, which are valid only for the Frobenius norm, the new bounds, as well as those in [6] for complex matrices, are extended to any unitarily invariant matrix norm. The bounds ...

In this paper we focus on the issue of normalization of the affinity matrix in spectral clustering. We show that the difference between N-cuts and Ratio-cuts is in the error measure being used (relative-entropy versus L1 norm) in finding the closest doubly-stochastic matrix to the input affinity matrix. We then develop a scheme for finding the optimal, under Frobenius norm, doubly-stochastic ap...

Methods to compute verified error bounds for the p-norm condition number of a matrix are discussed for p ∈ {1, 2,∞} and the Frobenius norm. We consider the cases of a real or complex, point or interval input matrix. In the latter case the condition number of all matrices within the interval matrix are bounded. A special method for extremely ill-conditioned matrices is derived as well. Numerical...

The stability radius of an n×n matrix A (or distance to instability) is a well-known measure of robustness of stability of the linear stable dynamical system ẋ = Ax. Such a distance is commonly measured either in the 2-norm or in the Frobenius norm. Even if the matrix A is real, the distance to instability is most often considered with respect to complex valued matrices (in such case the two no...

We estimate the rate of convergence and sample complexity of a recent robust estimator for a generalized version of the inverse covariance matrix. This estimator is used in a convex algorithm for robust subspace recovery (i.e., robust PCA). Our model assumes a sub-Gaussian underlying distribution and an i.i.d. sample from it. Our main result shows with high probability that the norm of the diff...

In this paper we are interested in computing linear least squares (LLS) condition numbers to measure the numerical sensitivity of an LLS solution to perturbations in data. We propose a statistical estimate for the normwise condition number of an LLS solution where perturbations on data are mesured using the Frobenius norm for matrices and the Euclidean norm for vectors. We also explain how cond...

We presented a new lower bound on minimal singular values of real matrices based on Frobenius norm and determinant and showed in [4] that under certain assumptions on the matrix is our estimate sharper than a recent lower bound from Hong and Pan [3]. In this paper we show, under which conditions is our lower bound sharper than two other recent lower bounds for minimal singular values based on a...

Abstract: Low-Rank Tensor Recovery (LRTR), the higher order generalization of Low-Rank Matrix Recovery (LRMR), is especially suitable for analyzing multi-linear data with gross corruptions, outliers and missing values, and it attracts broad attention in the fields of computer vision, machine learning and data mining. This paper considers a generalized model of LRTR and attempts to recover simul...

We provide quantitative bounds on the characterization of multiparticle separable states by states that have locally symmetric extensions. The bounds are derived from two-particle bounds and relate to recent studies on quantum versions of de Finetti's theorem. We discuss algorithmic applications of our results, in particular a quasipolynomial-time algorithm to decide whether a multiparticle qua...

In this paper we obtain the formula for computing the condition number of a complex matrix, which is related to the outer generalized inverse of a given matrix. We use the Schur decomposition of a matrix. We characterize the spectral norm and the Frobenius norm of the relative condition number of the generalized inverse, and the level-2 condition number of the generalized inverse. The sensitivi...