Optimal frame designs for multitasking devices with weight restrictions

Abstract

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=(Fopj)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 $ \sum \limits_{j\in \mathbb I_{m}}\text {P}_{\varphi }(\mathcal F_{j}^{\text {op}})\leq \sum \limits_{j\in \mathbb I_{m}}\text {P}_{\varphi }(\mathcal F_{j}) $ for every (α, d)-design Φ=(Fj)j∈Im, where Pφ(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, Fopj is a frame for Cdj, for j∈Im. This corresponds to the existence of optimal encoding-decoding schemes for multitasking devices with energy restrictions.