where B ∈ C, C ∈ C are fixed matrices and ∆ is an element of a subset ∆ of C. It is assumed that ∆ is closed, connected and contains the zero matrix. The size of a perturbation ∆ ∈ ∆ is measured by a norm ‖ · ‖ on C. (a) The structued pseudospectrum (also called spectral value set) of the triple (A,B,C) with respect to the perturbation class ∆, the underlying norm ‖ · ‖ and the perturbation lev...

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...

Abstract. Sharp error estimates in terms of the fractional Laplacian and a weighted Besov norm are obtained for Pitt’s inequality by using the spectral representation with weights for the fractional Laplacian due to Frank, Lieb and Seiringer and the sharp Stein-Weiss inequality. Dilation invariance, group symmetry on a non-unimodular group and a nonlinear Stein-Weiss lemma are used to provide s...

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 provide a unified variational inequality framework for the study of fundamental properties of the Nash equilibrium in network games. We identify several conditions on the underlying network (in terms of spectral norm, infinity norm and minimum eigenvalue of its adjacency matrix) that guarantee existence, uniqueness, convergence and continuity of equilibrium in general network games with mult...

We use induction and interpolation techniques to prove that a composition operator induced by a map φ is bounded on the weighted Bergman space Aα(H) of the right half-plane if and only if φ fixes ∞ non-tangentially, and has a finite angular derivative λ there. We further prove that in this case the norm, essential norm, and spectral radius of the operator are all equal, and given by λ.

In this paper, a new simple but yet efficient spectral expression of the frequency-limited H2-norm, denoted H2,ω-norm, is introduced. The proposed new formulation requires the computation of the system eigenvalues and eigenvectors only, and provides thus an alternative to the well established Gramian-based approach. The interest of this new formulation is in three-folds: (i) it provides a new t...

We define the complete numerical radius norm for homomorphisms from any operator algebra into B(H), and show that this norm can be computed explicitly in terms of the completely bounded norm. This is used to show that if K is a complete Cspectral set for an operator T , then it is a complete M -numerical radius set, where M = 12 (C + C −1). In particular, in view of Crouzeix’s theorem, there is...