Non-orthogonal Fusion Frames and the Sparsity of Fusion Frame Operators

Abstract

Fusion frames have become a major tool in the implementation of distributed systems. The effectiveness of fusion frame applications in distributed systems is reflected in the efficiency of the end fusion process. This in turn is reflected in the efficiency of the inversion of the fusion frame operator S W , which in turn is heavily dependent on the sparsity of S W . We will show that sparsity of the fusion frame operator naturally exists by introducing a notion of non-orthogonal fusion frames. We show that for a fusion frame {W i , v i } i∈I , if dim(W i ) = k i , then the matrix of the non-orthogonal fusion frame operator S W has in its corresponding location at most a k i × k i block matrix. We provide necessary and sufficient conditions for which the new fusion frame operator S W is diagonal and/or a multiple of an identity. A set of other critical questions are also addressed. A scheme of multiple fusion frames whose corresponding fusion frame operator becomes an diagonal operator is also examined.

Tight and random nonorthogonal fusion frames

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This paper continues the investigation of nonorthogonal fusion frames started in [7]. First we show that tight nonorthogonal fusion frames a relatively easy to com by.

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“…Typically we think of the dimension of each subspaces W i as being much smaller than the dimension of H so that a high dimensional signal f can be reconstructed from several low dimensional measurements {π i f } m i=1 . In [7], we introduced the idea of nonorthogonal fusion frames in order to achieve sparsity of the fusion frame operator. The basic observation in [7] is that replacing orthogonal projections π i in the original definition of fusion frames [5] by nonorthogonal projections P i onto the same subspaces W i can result in a fusion frame operator which is much sparser.…”

“…In [7], we introduced the idea of nonorthogonal fusion frames in order to achieve sparsity of the fusion frame operator. The basic observation in [7] is that replacing orthogonal projections π i in the original definition of fusion frames [5] by nonorthogonal projections P i onto the same subspaces W i can result in a fusion frame operator which is much sparser. This is because, for example, one can always choose the null space of the projection P i to contain some basis elements {e ij } j that are complementary to the subspace W i , and thereby nullify some columns of P i , which in turn results in sparsity of the (new) fusion frame operator.…”

Density Theorems for Nonuniform Sampling of Bandlimited Functions Using Derivatives or Bunched Measurements

“…In the case of bunched sampling, we will first show the existence of a particular fusion frame [17,18]. We recall that a non-orthogonal fusion frame [16] for a Hilbert space H is a set of positive scalars {v n } n∈I and non-orthogonal projections {P n } n∈I , each with closed range, satisfying…”

Randomized Subspace Actions and Fusion Frames

“…where P W denotes the orthogonal projection onto a subspace W ⊂ H. If A = B then the fusion frame is said to be tight. Note that in the special case when each W n is a onedimensional subspace with W n = span(ϕ n ) and v n = ϕ n , the fusion frame inequality (1.3) reduces to the statement (1.1) that {ϕ n } n∈I is a frame for H. For further background on fusion frames see [6,7,3,5]. If {(W n , v n )} n∈I is a fusion frame for H then the associated fusion frame operator S : H → H is defined by…”