Optimal frame designs for multitasking devices with weight restrictions

BENAC, MARÍA J.; MASSEY, PEDRO; RUIZ, MARIANO; STOJANOFF, DEMETRIO

ADVANCES IN COMPUTATIONAL MATHEMATICS

SPRINGER

Año: 2020 vol. 46

1019-7168

Let d=(dj)j∈Im∈ℕm be a finite sequence (of dimensions) and α=(αi)i∈In be a sequence of positive numbers (of weights), where Ik= { 1 , ? , k} for k∈ ℕ. We introduce the (α, d)-designs, i.e., m-tuples Φ=(Fj)j∈Im such that Fj={fij}i∈In is a finite sequence in ℂdj, j∈ Im, and such that the sequence of non-negative numbers (∥fij∥2)j∈Im forms a partition of αi, i∈ In. We characterize the existence of (α, d)-designs with prescribed properties in terms of majorization relations. We show, by means of a finite step algorithm, that there exist (α, d)-designs Φop=(Fjop)j∈Im that are universally optimal; that is, for every convex function φ: [0 , ∞) → [0 , ∞) , then Φop minimizes the joint convex potential induced by φ among (α, d)-designs, namely∑j∈ImPφ(Fjop)≤∑j∈ImPφ(Fj) for every (α, d)-design Φ=(Fj)j∈Im, where P φ(F) = tr (φ(SF)) ; in particular, Φop minimizes both the joint frame potential and the joint mean square error among (α, d)-designs. We show that in this case, Fjop is a frame for ℂdj, for j∈ Im. This corresponds to the existence of optimal encoding-decoding schemes for multitasking devices with energy restrictions.