The spectral k-support norm enjoys good estimation properties in low rank matrix learning problems, empirically outperforming the trace norm. Its unit ball is the convex hull of rank k matrices with unit Frobenius norm. In this paper we generalize the norm to the spectral (k, p)support norm, whose additional parameter p can be used to tailor the norm to the decay of the spectrum of the underlyi...

This paper studies matrix completion under a general sampling model using the max-norm as a convex relaxation for the rank of the matrix. The optimal rate of convergence is established for the Frobenius norm loss. It is shown that the max-norm constrained minimization method is rate-optimal and it yields a more stable approximate recovery guarantee, with respect to the sampling distributions, t...

We consider the estimation of the Brain Electrical Sources (BES) matrix from noisy electroencephalographic (EEG) measurements, commonly named as the EEG inverse problem. We propose a new method to induce neurophysiological meaningful solutions, which takes into account the smoothness, structured sparsity, and low rank of the BES matrix. The method is based on the factorization of the BES matrix...

it is well known that if the coefficient matrix in a linear system is large and sparse or sometimes not readily available, then iterative solvers may become the only choice. the block solvers are an attractive class of iterative solvers for solving linear systems with multiple right-hand sides. in general, the block solvers are more suitable for dense systems with preconditioner. in this paper,...

We show that given an estimate  that is close to a general high-rank positive semidefinite (PSD) matrix A in spectral norm (i.e., ‖Â−A‖2 ≤ δ), the simple truncated Singular Value Decomposition of  produces a multiplicative approximation of A in Frobenius norm. This observation leads to many interesting results on general high-rank matrix estimation problems: 1. High-rank matrix completion: we...

In the context of Krylov methods for solving systems of linear equations, expressions and bounds are derived for the norm of the minimal residual, like the one produced by GMRES or MINRES. It is shown that the minimal residual norm is large as long as the Krylov basis is well-conditioned. In the context of non-normal matrices, examples are given where the minimal residual norm is a function of ...

The nuclear norm (sum of singular values) of a matrix is often used in convex heuristics for rank minimization problems in control, signal processing, and statistics. Such heuristics can be viewed as extensions of l1-norm minimization techniques for cardinality minimization and sparse signal estimation. In this paper we consider the problem of minimizing the nuclear norm of an affine matrix val...

An iterative algorithm is constructed to solve the generalized coupled Sylvester matrix equations AXB − CYD,EXF − GYH M,N , which includes Sylvester and Lyapunov matrix equations as special cases, over generalized reflexive matrices X and Y . When the matrix equations are consistent, for any initial generalized reflexive matrix pair X1, Y1 , the generalized reflexive solutions can be obtained b...

In the present paper‎, ‎we propose an iterative algorithm for‎ ‎solving the generalized $(P,Q)$-reflexive solution of the quaternion matrix‎ ‎equation $overset{u}{underset{l=1}{sum}}A_{l}XB_{l}+overset{v} ‎{underset{s=1}{sum}}C_{s}widetilde{X}D_{s}=F$‎. ‎By this iterative algorithm‎, ‎the solvability of the problem can be determined automatically‎. ‎When the‎ ‎matrix equation is consistent over...

this paper is pertained with the problem of admissibility analysis of uncertain discrete-time nonlinear singular systems by adopting the state-space takagi-sugeno fuzzy model with time-delays and norm-bounded parameter uncertainties. lyapunov krasovskii functionals are constructed to obtain delay-dependent stability condition in terms of linear matrix inequalities, which is dependent on the low...