In this paper, we present a novel Frobenius Norm Filter (FNF), which is a spatially selective noise filtration technique in the wavelet subband domain. We address the issue of denoising of images corrupted with additive, multiplicative, and uncorrelated noise. The proposed nonlinear filter is an adaptive order statistic filter functioning on the L^2 space, which modulates itself according to th...

We propose an iterative algorithm for solving the reflexive solution of the quaternion matrix equation AXB + CXHD = F. When the matrix equation is consistent over reflexive matrix X, a reflexive solution can be obtained within finite iteration steps in the absence of roundoff errors. By the proposed iterative algorithm, the least Frobenius norm reflexive solution of the matrix equation can be d...

–The present work relates to a novel content based image authentication frame work. Here we take statistical based four methods for creating watermark. All these methods embed that statistical feature into the host image. In first method, Frobenius Norm of the host image is taken as watermark using ICA technique. The other methods used to create watermarks are namely, mean, standard deviation a...

We study the problem of matrix estimation and matrix completion under a general framework. This framework includes several important models as special cases such as the gaussian mixture model, mixed membership model, bi-clustering model and dictionary learning. We consider the optimal convergence rates in a minimax sense for estimation of the signal matrix under the Frobenius norm and under the...

We numerically construct slowly relaxing local operators in a nonintegrable spin-1/2 chain. Restricting the support of the operator to M consecutive spins along the chain, we exhaustively search for the operator that minimizes the Frobenius norm of the commutator with the Hamiltonian. We first show that the Frobenius norm bounds the time scale of relaxation of the operator at high temperatures....

Several recent randomized linear algebra algorithms rely upon fast dimension reduction methods. A popular choice is the Subsampled Randomized Hadamard Transform (SRHT). In this article, we address the efficacy, in the Frobenius and spectral norms, of an SRHT-based low-rank matrix approximation technique introduced by Woolfe, Liberty, Rohklin, and Tygert. We establish a slightly better Frobenius...

We present a generalization bound for feedforward neural networks with ReLU activations in terms of the product of the spectral norm of the layers and the Frobenius norm of the weights. The key ingredient is a bound on the changes in the output of a network with respect to perturbation of its weights, thereby bounding the sharpness of the network. We combine this perturbation bound with the PAC...

Today, we will continue with our discussion of scalar and matrix concentration, with a discussion of the matrix analogues of Markov’s, Chebychev’s, and Chernoff’s Inequalities. Then, we will return to bounding the error for our approximating matrix multiplication algorithm. We will start with using Hoeffding-Azuma bounds from last class to get improved Frobenius norm bounds, and then (next time...

Disclaimer: These notes have not been subjected to the usual scrutiny reserved for formal publications. In this lecture we describe applications of low rank approximation in optimization. Firstly, let us give a short overview of the last lecture. We defined the operator norm of a matrix ‖.‖2 and the Frobenius norm ‖.‖F and we showed that the best rank k approximation of a given matrix M is the ...