We investigate the generalizability of deep learning based on the sensitivity to input perturbation. We hypothesize that the high sensitivity to the perturbation of data degrades the performance on it. To reduce the sensitivity to perturbation, we propose a simple and effective regularization method, referred to as spectral norm regularization, which penalizes the high spectral norm of weight m...

Given four complex matrices $A$‎, ‎$B$‎, ‎$C$ and $D$ where $Ainmathbb{C}^{ntimes n}$‎ ‎and $Dinmathbb{C}^{mtimes m}$ and let the matrix $left(begin{array}{cc}‎ A & B ‎ C & D‎ end{array} right)$ be a normal matrix and‎ assume that $lambda$ is a given complex number‎ ‎that is not eigenvalue of matrix $A$‎. ‎We present a method to calculate the distance norm (with respect to 2-norm) from $D$‎ to ...

We find an expression for the Gateaux derivative of C⁎-algebra norm. Using this, we obtain a characterization orthogonality operator A∈B(H,K) to subspace, under assumption dist(A,K(H,K))<‖A‖. subdifferential set norm function at A∈B(H) when dist(A,K(H))<‖A‖. also give new proofs known results on closely related notions smooth points and Birkhoff-James spaces B(H) Cb(Ω), respectively.

RO-algebras are defined and studied. For RO-algebra T , using properties of partial order, it is established that the set of bounded elements can be endowed with C-norm. The structure of commutative subalgebras of T is considered and the Spectral Theorem for any self-adjoint element of T is proven.

Developed in 1999 by Akemann, Anderson, and Weaver, the spectral scale of an n × n matrix A, is a convex, compact subset of R that reveals important spectral information about A [4]. In this paper we present new information found in the spectral scale of a matrix. Given a matrix A = A1 + iA2 with A1 and A2 self-adjoint and A2 6= 0, we show that faces in the boundary of the spectral scale of A t...

In this work we solve the problem of the minimization of the spectral norm of the SOR operator associated with a block two-cyclic consistently ordered matrix A ∈ C, assuming that the corresponding Jacobi matrix has eigenvalues μ ∈ [−β, β] ∪ [−ıα, ıα], with β ∈ [0, 1), α ∈ [0,+∞) and ı = √ −1. Previous results obtained by other researchers are extended.

We develop and analyze a first-order system least-squares spectral method for the second-order elliptic boundary value problem with variable coefficients. We first analyze the Chebyshev weighted norm least-squares functional defined by the sum of the Lwand H−1 w norm of the residual equations and then we replace the negative norm by the discrete negative norm and analyze the discrete Chebyshev ...

A function f : F2 → {−1, 1} is called linear-isomorphic to g if f = g ◦ A for some non-singular matrix A. In the g-isomorphism problem, we want a randomized algorithm that distinguishes whether an input function f is linear-isomorphic to g or far from being so. We show that the query complexity to test g-isomorphism is essentially determined by the spectral norm of g. That is, if g is close to ...