We investigate necessary and sufficient conditions for a nonexpansive map $f$ on Banach space $X$ to have surjective displacement, that is, $f - \mathrm{id}$ onto $X$. In particular, we give computable condition when is finite dimensional with polyhedral norm. similar fixed point of norm be unique. also consider applications nonlinear Perron-Frobenius theory suggest some additional displacement...

This brief develops two algorithms for the order reduction of two-dimensional (2-D) FIR digital filters, which lead to two indirect design methods. Specifically, we show how an optimal L2 design can be derived as a matrix approximation problem in the Frobenius norm, and how a suboptimal L1 design can be obtained by solving a matrix approximation problem in the 2-norm using the Davis–Kahan–Weinb...

We study a simple two step procedure for estimating sparse precision matrices from data with missing values, which is tractable in high-dimensions and does not require imputation of the missing values. We provide rates of convergence for this estimator in the spectral norm, Frobenius norm and element-wise `∞ norm. Simulation studies show that this estimator compares favorably with the EM algori...

We consider in this paper the problem of noisy 1-bit matrix completion under a general non-uniform sampling distribution using the max-norm as a convex relaxation for the rank. A max-norm constrained maximum likelihood estimate is introduced and studied. The rate of convergence for the estimate is obtained. Information-theoretical methods are used to establish a minimax lower bound under the ge...

We give dimension-free and data-dependent bounds for linear multi-task learning where a common linear operator is chosen to preprocess data for a vector of task specific linear-thresholding classifiers. The complexity penalty of multi-task learning is bounded by a simple expression involving the margins of the task-specific classifiers, the Hilbert-Schmidt norm of the selected preprocessor and ...

In this paper we get the explicit condition number formulas for the W–weighted Drazin inverse of a rectangular matrix using the Schur decomposition and the spectral norm. We characterize the spectral norm and the Frobenius norm of the relative condition number of the W– weighted Drazin inverse, and the level-2 condition number of the W– weighted Drazin inverse. The sensitivity for the W–weighte...

The operator norm of an n × n complex matrix A is its norm as a linear operator on the Euclidean space I C; i.e., ‖A‖ = sup{‖Ax‖ : x ∈ I C, ‖x‖ = 1}. The Frobenius norm of A is defined as ‖A‖2 = ( ∑ i,j |aij|) 1 2 = (trA∗A) 1 2 . Both these norms are used frequently in analysis of matrices. They can also be described in terms of the singular values of A − the square roots of the eigenvalues of ...

We present new examples illustrating the fact that optimal Hankel structured rank-1 approximation of matrices is usually different for Frobenius and spectral norm. Further, we compare our results to well-known Cadzow algorithm [3].

We study a simple two step procedure for estimating sparse precision matrices from data with missing values, which is tractable in high-dimensions and does not require imputation of the missing values. We provide rates of convergence for this estimator in the spectral norm, Frobenius norm and element-wise `∞ norm. Simulation studies show that this estimator compares favorably with the EM algori...

In this work, we propose a new randomized algorithm for computing a low-rank approximation to a given matrix. Taking an approach different from existing literature, our method first involves a specific biased sampling, with an element being chosen based on the leverage scores of its row and column, and then involves weighted alternating minimization over the factored form of the intended low-ra...