We answer a number of open problems in frame theory concerning the decomposition of frames into linearly independent and/or spanning sets. We prove that Parseval frames with norms bounded away from 1 can be decomposed into a number of sets whose complements are spanning, where the number of these sets only depends on the norm bound. Further, we prove a stronger result for Parseval frames whose ...

In this paper, we study the finite Gabor system for oversampling schemes. A characterization of dual tight frame using discrete time Zak transform is given. Also, a method to calculate coefficients expansion in case and necessary sufficient condition existence biorthogonal pair Riesz basis l2(ZL) Further, introduce notion block sliding Fourier (BSDFT) which reduces computational complexity give...

We introduce Parseval networks, a form of deep neural networks in which the Lipschitz constant of linear, convolutional and aggregation layers is constrained to be smaller than 1. Parseval networks are empirically and theoretically motivated by an analysis of the robustness of the predictions made by deep neural networks when their input is subject to an adversarial perturbation. The most impor...

We introduce Parseval networks, a form of deep neural networks in which the Lipschitz constant of linear, convolutional and aggregation layers is constrained to be smaller than 1. Parseval networks are empirically and theoretically motivated by an analysis of the robustness of the predictions made by deep neural networks when their input is subject to an adversarial perturbation. The most impor...

Phase retrieval has become a very active area of research. We will classify when phase retrieval by Parseval frames passes to the Naimark complement and when phase retrieval by projections passes to the orthogonal complements. We introduce a new concept we call norm retrieval and show that this is what is necessary for passing phase retrieval to complements. This leads to a detailed study of no...

We make a detailed study of norm retrieval. We give several classification theorems for norm retrieval and give a large number of examples to go with the theory. One consequence is a new result about Parseval frames: If a Parseval frame is divided into two subsets with spans W1, W2 and W1 ∩W2 = {0}, then W1 ⊥W2.

We examine relations between popular variational methods in image processing and classical operator splitting methods in convex analysis. We focus on a gradient descent reprojection algorithm for image denoising and the recently proposed Split Bregman and alternating Split Bregman methods. By identifying the latter with the so-called DouglasRachford splitting algorithm we can guarantee its conv...