Quantized Frame Expansions with Erasures

Frames have been used to capture significant signal characteristics, provide numerical stability of reconstruction, and enhance resilience to additive noise. This paper places frames in a new setting, where some of the elements are deleted. Since proper subsets of frames are sometimes themselves frames, a quantized frame expansion can be a useful representation even when some transform coefficients are lost in transmission. This yields robustness to losses in packet networks such as the Internet. With a simple model for quantization error, it is shown that a normalized frame minimizes mean-squared error if and only if it is tight. With one coefficient erased, a tight frame is again optimal among normalized frames, both in average and worst-case scenarios. For more erasures, a general analysis indicates some optimal designs. Being left with a tight frame after erasures minimizes distortion, but considering also the transmission rate and possible erasure events complicates optimizations greatly.     

Equal-Norm Tight Frames with Erasures

Equal-norm tight frames have been shown to be useful for robust data transmission. The losses in the network are modeled as erasures of transmitted frame coefficients. We give the first systematic study of the general class of equal-norm tight frames and their properties. We search for efficient constructions of such frames. We show that the only equal-norm tight frames with the group structure and one or two generators are the generalized harmonic frames. Finally, we give a complete classification of frames in terms of their robustness to erasures.     

The Structure of Minimizers of the Frame Potential on Fusion Frames

In this paper we study the fusion frame potential that is a generalization of the Benedetto-Fickus (vectorial) frame potential to the finite-dimensional fusion frame setting. We study the structure of local and global minimizers of this potential, when restricted to suitable sets of fusion frames. These minimizers are related to tight fusion frames as in the classical vector frame case. Still, tight fusion frames are not as frequent as tight frames; indeed we show that there are choices of parameters involved in fusion frames for which no tight fusion frame can exist. We exhibit necessary and sufficient conditions for the existence of tight fusion frames with prescribed parameters, involving the so-called Horn-Klyachko’s compatibility inequalities. The second part of the work is devoted to the study of the minimization of the fusion frame potential on a fixed sequence of subspaces, with a varying sequence of weights. We related this problem to the index of the Hadamard product by positive matrices and use it to give different characterizations of these minima.     

New characterizations of fusion frames (frames of subspaces)

In this article, we give new characterizations of fusion frames, on the properties of their synthesis operators, on the behavior of fusion frames under bounded operators with closed range, and on erasures of subspaces of fusion frames. Furthermore we show that every fusion frame is the image of an orthonormal fusion basis under a bounded surjective operator.     

Random Matrices and Erasure Robust Frames

Data erasure can often occur in communication. Guarding against erasures involves redundancy in data representation. Mathematically this may be achieved by redundancy through the use of frames. One way to measure the robustness of a frame against erasures is to examine the worst case condition number of the frame with a certain number of vectors erased from the frame. The term numerically erasure-robust frames was introduced in Fickus and Mixon (Linear Algebra Appl 437:1394–1407, 2012) to give a more precise characterization of erasure robustness of frames. In the paper the authors established that random frames whose entries are drawn independently from the standard normal distribution can be robust against up to approximately 15 % erasures, and asked whether there exist frames that are robust against erasures of more than 50 %. In this paper we show that with very high probability random frames are, independent of the dimension, robust against erasures as long as the number of remaining vectors is at least \(1+\delta _0\) times the dimension for some \(\delta _0>0\). This is the best possible result, and it also implies that the proportion of erasures can be arbitrarily close to 1 while still maintaining robustness. Our result depends crucially on a new estimate for the smallest singular value of a rectangular random matrix with independent standard normal entries.     

Numerically erasure-robust frames

Given a channel with additive noise and adversarial erasures, the task is to design a frame that allows for stable signal reconstruction from transmitted frame coefficients. To meet these specifications, we introduce numerically erasure-robust frames. We first consider a variety of constructions, including random frames, equiangular tight frames and group frames. Later, we show that arbitrarily large erasure rates necessarily induce numerical instability in signal reconstruction. We conclude with a few observations, including some implications for maximal equiangular tight frames and sparse frames.     

Minimizing Fusion Frame Potential

Fusion frames are an emerging topic of frame theory, with applications to encoding and distributed sensing. However, little is known about the existence of tight fusion frames. In traditional frame theory, one method for showing that unit norm tight frames exist is to characterize them as the minimizers of an energy functional, known as the frame potential. We generalize the frame potential to the fusion frame setting. In particular, we introduce the fusion frame potential, and show how its minimization is equivalent to the minimization of the traditional frame potential over a particular domain. We then study this minimization problem in detail. Specifically, we show that if the desired number of fusion frame subspaces is large, and if the desired dimension of these subspaces is small compared to the dimension of the underlying space, then a tight fusion frame of those dimensions will necessarily exist, being a minimizer of the fusion frame potential.     

Sparse fusion frames: existence and construction

Fusion frame theory is an emerging mathematical theory that provides a natural framework for performing hierarchical data processing. A fusion frame can be regarded as a frame-like collection of subspaces in a Hilbert space, and thereby generalizes the concept of a frame for signal representation. However, when the signal and/or subspace dimensions are large, the decomposition of the signal into its fusion frame measurements through subspace projections typically requires a large number of additions and multiplications, and this makes the decomposition intractable in applications with limited computing budget. To address this problem, in this paper, we introduce the notion of a sparse fusion frame, that is, a fusion frame whose subspaces are generated by orthonormal basis vectors that are sparse in a ‘uniform basis’ over all subspaces, thereby enabling low-complexity fusion frame decompositions. We study the existence and construction of sparse fusion frames, but our focus is on developing simple algorithmic constructions that can easily be adopted in practice to produce sparse fusion frames with desired (given) operators. By a desired (or given) operator we simply mean one that has a desired (or given) set of eigenvalues for the fusion frame operator. We start by presenting a complete characterization of Parseval fusion frames in terms of the existence of special isometries defined on an encompassing Hilbert space. We then introduce two general methodologies to generate new fusion frames from existing ones, namely the Spatial Complement Method and the Naimark Complement Method, and analyze the relationship between the parameters of the original and the new fusion frame. We proceed by establishing existence conditions for 2-sparse fusion frames for any given fusion frame operator, for which the eigenvalues are greater than or equal to two. We then provide an easily implementable algorithm for computing such 2-sparse fusion frames.     

Complex equiangular tight frames and erasures

In this paper we demonstrate that there are distinct differences between real and complex equiangular tight frames (ETFs) with regards to erasures. For example, we prove that there exist arbitrarily large non-trivial complex equiangular tight frames which are optimal against three erasures, and that such frames come from a unique class of complex ETFs. In addition, we extend certain results in Bodmann and Paulsen (2005) [2] to complex vector spaces as well as show that other results regarding real ETFs are not valid for complex ETFs.     

Full Spark Frames

Finite frame theory has a number of real-world applications. In applications like sparse signal processing, data transmission with robustness to erasures, and reconstruction without phase, there is a pressing need for deterministic constructions of frames with the following property: every size-M subcollection of the M-dimensional frame elements is a spanning set. Such frames are called full spark frames, and this paper provides new constructions using the discrete Fourier transform. Later, we prove that full spark Parseval frames are dense in the entire set of Parseval frames, meaning full spark frames are abundant, even if one imposes an additional tightness constraint. Finally, we prove that testing whether a given matrix is full spark is hard for NP under randomized polynomial-time reductions, indicating that deterministic full spark constructions are particularly significant because they guarantee a property which is otherwise difficult to check.     

<Emphasis Type="Italic">K</Emphasis>-fusion Frames and the Corresponding Generators for Unitary Systems

Motivated by K-frames and fusion frames, we study K-fusion frames in Hilbert spaces. By the means of operator K, frame operators and quotient operators, several necessary and sufficient conditions for a sequence of closed subspaces and weights to be a K-fusion frame are obtained, and operators preserving K-fusion frames are discussed. In particular, we are interested in the K-fusion frames with the structure of unitary systems. Given a unitary system which has a complete wandering subspace, we give a necessary and sufficient condition for a closed subspace to be a K-fusion frame generator.     

Frames, modular functions for shift-invariant subspaces and FMRA wavelet frames

We introduce the concept of the modular function for a shift-invariant subspace that can be represented by normalized tight frame generators for the shift-invariant subspace and prove that it is independent of the selections of the frame generators for the subspace. We shall apply it to study the connections between the dimension functions of wavelet frames for any expansive integer matrix and the multiplicity functions for general multiresolution analysis (GMRA). Given a frame mutiresolution analysis (FMRA), we show that the standard construction formula for orthonormal multiresolution analysis wavelets does not yield wavelet frames unless the underlying FMRA is an MRA. A modified explicit construction formula for FMRA wavelet frames is given in terms of the frame scaling functions and the low-pass filters.

     

Robustness and surgery of frames

We characterize frames in Rⁿ that are robust to k erasures. The characterization is given in terms of the support of the null space of the synthesis operator of the frame. A necessary and sufficient condition is given for when an (r, k)-surgery on unit-norm tight frames in R² are possible. Also a generalization of a known characterization of the existence of tight frames with prescribed norms is given.     

K-fusion Frames and the Corresponding Generators for Unitary Systems

Motivated by K-frames and fusion frames, we study K-fusion frames in Hilbert spaces.By the means of operator K, frame operators and quotient operators, several necessary and sufficient conditions for a sequence of closed subspaces and weights to be a K-fusion frame are obtained, and operators preserving K-fusion frames are discussed. In particular, we are interested in the K-fusion frames with the structure of unitary systems. Given a unitary system which has a complete wandering subspace, we give a necessary and sufficient condition for a closed subspace to be a K-fusion frame generator.     

Linearly connected sequences and spectrally optimal dual frames for erasures

In the case that a frame is prescribed for applications and erasures occur in the process of data transmissions, we examine optimal dual frames for the recovery from single erasures. In contrast to earlier papers, we consider the spectral radius of the error operator instead of its operator norm as a measure of optimality. This notion of optimality is natural when the Neumann series is used to recover the original data in an iterative manner. We obtain a complete characterization of spectrally one-erasure optimal dual frames in terms of the redundancy distribution of the prescribed frame. Our characterization relies on the connection between erasure optimal frames and the linear connectivity property of the frame. We prove that the linear connectivity property is equivalent to the intersection dependent property, and is also closely related to the well-known concept of a k-independent set. Additionally, we also establish several necessary and sufficient conditions for the existence of an alternate dual frame to make the iterative reconstruction work.     

Wavelet frames for (not necessarily reducing) affine subspaces II: The structure of affine subspaces

This is a continuation of the investigation into the theory of wavelet frames for general affine subspaces. The main focus of this paper is on the structural properties of affine subspaces. We show that every affine subspace is the orthogonal direct sum of at most three purely non-reducing subspaces, while every reducing subspace (with respect to the dilation and translation operators) is the orthogonal direct sum of two purely non-reducing ones. This result is obtained through considering the basic question as to when the orthogonal complement of an affine subspace in another one is still affine. Motivated by the fundamental question as to whether every affine subspace is singly-generated, and by a recent result that every singly generated purely non-reducing subspace admits a singly generated wavelet frame, we prove that every affine subspace can be decomposed into the direct sum of a singly generated affine subspace and some space of “small size”. As a consequence we establish a connection between the above mentioned two questions.     

Some properties of frames of subspaces obtained by operator theory methods

We study the relationship among operators, orthonormal basis of subspaces and frames of subspaces (also called fusion frames) for a separable Hilbert space H. We get sufficient conditions on an orthonormal basis of subspaces E={Ei}_i∈I of a Hilbert space K and a surjective T∈L(K,H) in order that {T(Ei)}_i∈I is a frame of subspaces with respect to a computable sequence of weights. We also obtain generalizations of results in [J.A. Antezana, G. Corach, M. Ruiz, D. Stojanoff, Oblique projections and frames, Proc. Amer. Math. Soc. 134 (2006) 1031-1037], which relate frames of subspaces (including the computation of their weights) and oblique projections. The notion of refinement of a fusion frame is defined and used to obtain results about the excess of such frames. We study the set of admissible weights for a generating sequence of subspaces. Several examples are given.     

Subspace Weyl-Heisenberg frames

A Weyl-Heisenberg frame (WH frame) for L²(ℝ) allows every square integrable function on the line to be decomposed into the infinite sum of linear combination of translated and modulated versions of a fixed function. Some sufficient conditions for g ∈ L²(ℝ) to be a subspace Weyl-Heisenberg frame were given in a recent work [3] by Casazza and Christensen. Obviously every invariant subspace (under translation and modulation) is cyclic if it has a subspace WH frame. In the present article we prove that the cyclicity property is also sufficient for a subspace to admit a WH frame. We also investigate the dilation property for subspace Weyl-Heisenberg frames and show that every normalized tight subspace WH frame can be dilated to a normalized tight WH frame which is “maximal” In other words, every subspace WH frame is the compression of a WH frame which cannot be dilated anymore within the L²(ℝ) space. Communicated by Hans G. Feichtinger     

Approximations for Gabor and wavelet frames

Let be a frame vector under the action of a collection of unitary operators . Motivated by the recent work of Frank, Paulsen and Tiballi and some application aspects of Gabor and wavelet frames, we consider the existence and uniqueness of the best approximation by normalized tight frame vectors. We prove that for any frame induced by a projective unitary representation for a countable discrete group, the best normalized tight frame (NTF) approximation exists and is unique. Therefore it applies to Gabor frames (including Gabor frames for subspaces) and frames induced by translation groups. Similar results hold for semi-orthogonal wavelet frames.