Optimal frame designs for multitasking devices with weight restrictions

Abstract

Let d=(d_j)_j∈I_m ∈ N^m be a finite sequence (of dimensions) and α=(α_i)_i∈ I_n be a sequence of positive numbers (of weights), where I_k={1,...,k} for k ∈ N. We introduce the (α , d)-designs i.e., m-tuples Φ=( F_j)_j ∈ I_m such that F_j={ f_ij}_i∈ I_n is a finite sequence in C^{d_j}, j ∈ I_m, and such that the sequence of non-negative numbers (||f_ij||^2)_j ∈ I_m forms a partition of α_i, i ∈ I_n. 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 Φ^ op =(F_j^op)_j∈I_m that are universally optimal; that is, for every convex function φ:[0,∞)→ [0,∞) then Φ^ op minimizes the joint convex potential induced by φ among (α , d)-designs, namely Σ_{j ∈ I_m} P_φ( F_j^op) ≤ Σ_{j ∈ I_m} P_φ( F_j) for every (α , d)$-design Φ=( F_j)_{j∈ I_m}, where P_φ(F)=tr(φ(S_F)); in particular, Φ^ op minimizes both the joint frame potential and the joint mean square error among (α , d)-designs. We show that in this case F_j^op is a frame for C^{d_j}, for j ∈ I_m. This corresponds to the existence of optimal encoding-decoding schemes for multitasking devices with energy restrictions.