Procrustes Problems and Parseval Quasi-Dual Frames

Abstract

Parseval frames have particularly useful properties, and in some cases, they can be used to reconstruct signals which were analyzed by a non-Parseval frame. In this paper, we completely describe the degree to which such reconstruction is feasible. Indeed, notice that for fixed frames F and X with synthesis operators F and X, the operator norm of F X * − I measures the (normalized) worst-case error in the reconstruction of vectors when analyzed with X and synthesized with F . Hence, for any given frame F , we compute explicitly the infimum of the operator norms of F X * − I, where X is any Parseval frame. The X 's that minimize this quantity are called Parseval quasidual frames of F . Our treatment considers both finite and infinite Parseval quasi-dual frames.

Generalized frame operator distance problems

“…Hence, arguing as in the proof of Corollary 3.10, we conclude that this c satisfies Eq. (10). Note that c < λ 1 (S) because tr a = 0.…”

“…Still, some other norms are also of interest such as weighted norms, the p-norms for 1 ≤ p (that contain the Frobenius norm), or the more general class of unitarily invariant norms. Some of the most important choices for X are the set of: selfadjoint matrices, positive semidefinite matrices, correlation matrices, orthogonal projections, oblique projections, matrices with rank bounded by a fix number (see [10,11,13,15,24]).…”

Local extrema for Procustes problems in the set of positive definite matrices

“…Other norms that are also of interest are weighted norms, the p-norms for 1 ≤ p (which contains the Frobenius norm), or the more general class of unitarily invariant norms. With regards to the sets X , the most relevant choices are: selfadjoint matrices, positive semidefinite matrices, correlation matrices, orthogonal projections, oblique projections and matrices with rank bounded by a fix number (see for example [3,4,6,11,20]). In the special case of Procustes type problem, X is the unitary orbit of a positive (or selfadjoint) matrix B and A is also a positive (or selfadjoint) matrix.…”

Weighted least squares solutions of the equation AXB−C= 0

“…There are several examples of these minimization problems, in [11] a similar problem related with frame theory is studied, in [15] the existence of minimum of AX − I p in the finite dimensional setting is given. In [18], [17], [20], [21] and in [5], the existence of minimum of AX − C p in Hilbert spaces, with suitable hypotesis to warrant that AX − C ∈ S p , was studied using differentiation techniques and also in [14], where a connection between p-Schatten norms and the order in L(H) + (the cone of semidefinite positive operators) is established.…”