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Register a new userNew Constructions and Characterizations of Flat and Almost Flat Grassmannian Fusion Frames
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Authors
Emily J. King
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Configurations of subspaces like equichordal and equiisoclinic tight fusion frames, which are in some sense optimally spread apart and which also have reconstruction properties emulating those of orthonormal bases, are useful in various applications, such as wireless communications and quantum information theory. In this paper, a new construction of infinite classes of equichordal tight fusion frames built on semiregular divisible difference sets is presented. Sometimes this construction yields an equiisoclinic packing. Each of the constructed fusion frames is shown to have both a flat representation and a sparse representation. Furthermore, integrality conditions which characterize when equichordal and equiisoclinic fusion frames can have orthonormal bases with entries in a subring of the algebraic integers are proven. Keywords: fusion frame, Grassmannian packing, difference sets, simplex bound, equichordal, equiisoclinic
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We consider estimating a random vector from its noisy projections onto low dimensional subspaces constituting a fusion frame. A fusion frame is a collection of subspaces, for which the sum of the projection operators onto the subspaces is bounded below and above by constant multiples of the identity operator. We first determine the minimum mean-squared error (MSE) in linearly estimating the random vector of interest from its fusion frame projections, in the presence of white noise. We show that MSE assumes its minimum value when the fusion frame is tight. We then analyze the robustness of the constructed linear minimum MSE (LMMSE) estimator to erasures of the fusion frame subspaces. We prove that tight fusion frames consisting of equi-dimensional subspaces have maximum robustness (in the MSE sense) with respect to erasures of one subspace, and that the optimal subspace dimension depends on signal-to-noise ratio (SNR). We also prove that tight fusion frames consisting of equi-dimensional subspaces with equal pairwise chordal distances are most robust with respect to two and more subspace erasures. We call such fusion frames equi-distance tight fusion frames, and prove that the chordal distance between subspaces in such fusion frames meets the so-called simplex bound, and thereby establish connections between equi-distance tight fusion frames and optimal Grassmannian packings. Finally, we present several examples for construction of equi-distance tight fusion frames.
Fusion frame theory is an emerging mathematical theory that provides a natural framework for performing hierarchical data processing. A fusion frame is a frame-like collection of subspaces in a Hilbert space, thereby generalizing the concept of a frame for signal representation. In this paper, we study the existence and construction of fusion frames. We first present a complete characterization of a special class of fusion frames, called Parseval fusion frames. The value of Parseval fusion frames is that the inverse fusion frame operator is equal to the identity and therefore signal reconstruction can be performed with minimal complexity. We then introduce two general methods -- the spatial complement and the Naimark complement -- for constructing a new fusion frame from a given fusion frame. We then establish existence conditions for fusion frames with desired properties. In particular, we address the following question: Given $M, N, m in NN$ and ${lambda_j}_{j=1}^M$, does there exist a fusion frame in $RR^M$ with $N$ subspaces of dimension $m$ for which ${lambda_j}_{j=1}^M$ are the eigenvalues of the associated fusion frame operator? We address this problem by providing an algorithm which computes such a fusion frame for almost any collection of parameters $M, N, m in NN$ and ${lambda_j}_{j=1}^M$. Moreover, we show how this procedure can be applied, if subspaces are to be added to a given fusion frame to force it to become Parseval.
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The local structure of half conformally flat gradient Ricci almost solitons is investigated, showing that they are locally conformally flat in a neighborhood of any point where the gradient of the potential function is non-null. In opposition, if the gradient of the potential function is null, then the soliton is a steady traceless $kappa$-Einstein soliton and is realized on the cotangent bundle of an affine surface.
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