Equal-norm Tight Frames Research Articles

Equal-norm Parseval K-frames in Hilbert spaces with a new inequality

The focus of this paper is mainly on the frames of operators or K-frames on Hilbert spaces in Parseval cases. Since equal-norm tight frames play an important role for transmitting robust data, we aim to study this topic on Parseval K-frames. We find that each finite set of equal-norm of vectors can be extended to an equal-norm K-frame. We also find a correspondence between Parseval K-frames and the set of all closed subspaces of a finite Hilbert space. Furthermore, we provide a construction of dual equal-norm K-frames.

Maltsev equal-norm tight frames

A frame in $\mathbb{R}^d$ is a set of $n\geqslant d$ vectors whose linear span coincides with $\mathbb{R}^d$. A frame is said to be equal-norm if the norms of all its vectors are equal. Tight frames enable one to represent vectors in $\mathbb{R}^d$ in the form closest to the representation in an orthonormal basis. Every equal-norm tight frame is a useful tool for constructing efficient computational algorithms. The construction of such frames in $\mathbb{C}^d$ uses the matrix of the discrete Fourier transform, and the first constructions of equal-norm tight frames in $\mathbb{R}^d$ appeared only at the beginning of the 21st century. The present paper shows that Maltsev's note of 1947 was decades ahead of its time and turned out to be missed by the experts in frame theory, and Maltsev should be credited for the world's first design of an equal-norm tight frame in $\mathbb{R}^d$. Our main purpose is to show the historical significance of Maltsev's discovery. We consider his paper from the point of view of the modern theory of frames in finite-dimensional spaces. Using the Naimark projectors and other operator methods, we study important frame-theoretic properties of the Maltsev construction, such as the equality of moduli of pairwise scalar products (equiangularity) and the presence of full spark, that is, the linear independence of any subset of $d$ vectors in the frame.

Full spark frames in the orbit of a representation

We present a new infinite family of full spark equal norm tight frames in finite dimensions arising from a unitary group representation, where the underlying group is the semi-direct product of a cyclic group by a group of automorphisms. The only previously known algebraically constructed infinite families were the harmonic, Gabor and Dihedral frames. Our construction hinges on a theorem that requires no group structure. Additionally, we illustrate our results by providing explicit constructions of full spark frames.

Nice error frames, canonical abstract error groups and the construction of SICs

Nice error bases are generalisations of the Pauli matrices which have applications in quantum information theory. These orthonormal bases for the d×d matrices Md(C) also generalise the projective action of the Heisenberg group on Cd. Here we extend nice error bases to nice error frames. These are equal-norm tight frames for Md(C) consisting of d×d unitary matrices with a group indexing structure. We show that each nice error frame (irreducible faithful projective representation) is associated with a canonical abstract error group. This is calculated in number of examples, e.g., for all nice error bases for d<14, which then allows us to investigate which nice error bases might give rise to SICs (symmetric informationally complete positive operator valued measures). These results show that the current catalogue of nice error bases over counts. In particular, we give an explicit example of a SIC for d=6 with a nonabelian index group, and show that the Hoggar lines appear for various nice error bases, some of which are subgroups of the Clifford group. Thus all known SICs appear as orbits of subgroups of the Clifford group.

Polytopes of Eigensteps of Finite Equal Norm Tight Frames

Hilbert space frames generalize orthonormal bases to allow redundancy in representations of vectors while keeping good reconstruction properties. A frame comes with an associated frame operator encoding essential properties of the frame. We study a polytope that arises in an algorithm for constructing all finite frames with given lengths of frame vectors and spectrum of the frame operator, which is a Gelfand-Tsetlin polytope. For equal norm tight frames, we give a non-redundant description of the polytope in terms of equations and inequalities. From this we obtain the dimension and number of facets of the polytope.

On the construction of highly symmetric tight frames and complex polytopes

Many “highly symmetric” configurations of vectors in Cd, such as the vertices of the platonic solids and the regular complex polytopes, are equal-norm tight frames by virtue of being the orbit of the irreducible unitary action of their symmetry group. For nonabelian groups there are uncountably many such tight frames up to unitary equivalence. The aim of this paper is to single out those orbits which are particularly nice, such as those which are the vertices of a complex polytope. This is done by defining a finite class of tight frames of n vectors for Cd (n and d fixed) which we call the highly symmetric tight frames. We outline how these frames can be calculated from the representations of abstract groups using a computer algebra package. We give numerous examples, with a special emphasis on those obtained from the (Shephard–Todd) finite reflection groups. The interrelationships between these frames with complex polytopes, harmonic frames, equiangular tight frames, and Heisenberg frames (maximal sets of equiangular lines) are explored in detail.

Frame-Theoretic Analysis of Robust Filter Bank Frames to Quantization and Erasures

This paper presents the theoretic analysis for the robustness of filter bank (FB) frames in infinite dimensional Hilbert space l2(\BBZ) to quantization and erasures as well as studies the design of such robust frames, from the perspective of both frame and FB theory. First, a characterization of the eigenstructure for the frame operator and the induced Gram matrix of general FB frames is presented. The robustness of FB frames to erasures is investigated in detail, especially on the necessary and sufficient condition. Maximally robust frames are further analyzed by explicitly constructive methods. Moreover, the stability and even the possible FIR reconstruction of the subframe by the pseudoinverse are studied thoroughly. Next, we examine the optimal quantized FB frames. Introducing a novel notion named as frame energy, the universal optimality of tight FB frames to quantization is established, in contrast to the optimality of only equal norm tight frames shown in previous works. The added design freedom obtained by removing the equal norm constraint is explained and illustrated with examples. Finally, the effect of erasures on the quantized FB frames is studied by incorporating the probability of erasures, which shows the universal optimality of uniform tight FB frames for one erasure. The optimal FB frame is usually not of equal norm and actually its norm distribution follows the reverse waterfilling principle. An example of two-state erasure channel model is shown to further explain our result, followed by an analysis of the successive reconstruction which shows its potential application to frame-based multiple description coding.

On computing all harmonic frames of n vectors in [formula omitted

There are a finite number of inequivalent isometric frames (equal-norm tight frames) of n vectors for C d which are generated from a single vector by applying an Abelian group G of symmetries. Each of these so-called harmonic frames can be obtained by taking d rows of the character table of G; often in many different ways, which may even include using different Abelian groups. Using an algorithm implemented in the algebra package Magma, we determine which of these are equivalent. The resulting list of all harmonic frames for various choices of n and d is freely available, and it includes many properties of the frames such as: a simple description, which Abelian groups generate it, identification of the full group of symmetries, the minimum, average and maximum distance between vectors in the frame, and whether it is real or complex, lifted or unlifted. Additional attributes aimed at specific applications include: a measure of the cross correlation (Grassmannian frames), the number of erasures (robustness to erasures), and the diversity product of the full group of its symmetries (multiple-antenna code design). Some outstanding frames are identified and discussed, and a number of questions are answered by considering the examples on the list.

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.