Optimal dual frames and frame completions for majorization

Abstract

and IAM-CONICET Dedicated to the memory of "el flaco" L. A. Spinetta. AbstractIn this paper we consider two problems in frame theory. On the one hand, given a set of vectors F we describe the spectral and geometrical structure of optimal completions of F by a finite family of vectors with prescribed norms, where optimality is measured with respect to majorization. In particular, these optimal completions are the minimizers of a family of convex functionals that include the mean square error and the Benedetto-Fickus' frame potential. On the other hand, given a fixed frame F we describe explicitly the spectral and geometrical structure of optimal frames G that are in duality with F and such that the Frobenius norms of their analysis operators is bounded from below by a fixed constant. In this case, optimality is measured with respect to submajorization of the frames operators. Our approach relies on the description of the spectral and geometrical structure of matrices that minimize submajorization on sets that are naturally associated with the problems above.Given m ∈ N we denote by I m = {1, . . . , m} ⊆ N and 1 = 1 m ∈ R m denotes the vector with all its entries equal to 1. For a vector x ∈ R m we denote by x ↓ the rearrangement of x in decreasing order, and R m ↓ = {x ∈ R m : x = x ↓ } the set of ordered vectors.Given H ∼ = C d and K ∼ = C n , we denote by L(H , K) the space of linear operators

Construction of robust frames in erasure recovery

“…Some researchers studied the approximation of f directly by i∈Λ c c iφi , for which the error operator is E Λ = i∈Λ ϕ i ⊗φ i . There are a lot of works on how to minimize the approximation error, e.g., see [1][2][3][4][5][6][11][12][13][14][15][16][17].…”

Optimal frame designs for multitasking devices with weight restrictions

“…The first results were obtained for the frame potential in [4] and in a more general context in [9]. The case of general convex potentials was studied in [17,18,21,22,23,24,25,26] (in some cases in the more general setting of frame completion problems with prescribed norms).…”

“…Theorem 2.6 ([9, 23,24,25]). Let α = (α i ) i∈In ∈ (R n ≥0 ) ↓ and let d ∈ N be such that d ≤ n. Then, there exists γ op…”

“…It is then natural to wonder whether there are tight frames with norms prescribed by α. This question has motivated the study of the frame design problem (see [1,8,10,11,14,15,16,20] and [17,18,22,21,24,25,26] for the more general frame completion problem with prescribed norms). It is well known that in some cases there are no tight frames in the class of sequences in C d with norms prescribed by α; in these cases, it is natural to consider minimizers of the frame potential within this class, since the eigenvalues of the frame operator of such minimizers have minimal spread (thus, inducing more stable linear reconstruction processes).…”

Procrustes Problems and Parseval Quasi-Dual Frames

“…In this case we say that (F, G) is a dual pair of frames for H. It is worth pointing out that the canonical dual frame F # corresponds to the adjoint of the Moore-Penrose pseudoinverse of F under the previous bijection; this fact indicates that F # plays a key role among the set of all dual frames for F. Nevertheless, given a redundant frame F then the canonical dual frame F # is not always the best choice for a dual of F. For example, numerical stability comes into play when dealing with reconstruction formulas derived from a dual pair (F, G); in this case a measure of stability of the reconstruction algorithm for fixed F is given by the condition number of the frame operator S G of G. It is known [22] that the dual frames G that minimize this condition number are different (in general) from the canonical dual (see [4,20,26] for other examples of this phenomena). In this vein, D. Han [13] characterized those frames F for which there exists a dual frame for F, denoted by X = {x i } i∈N , which is a Parseval frame i.e.…”