Optimal frame designs for multitasking devices with weight restrictions

MARIANO RUIZ; PEDRO MASSEY; PEDRO MASSEY; MARÍA JOSÉ BENAC; DEMETRIO STOJANOFF; MARÍA JOSÉ BENAC; DEMETRIO STOJANOFF; MARIANO RUIZ

ADVANCES IN COMPUTATIONAL MATHEMATICS

SPRINGER

Año: 2020 vol. 46

1019-7168

Let d=(d_j)_{j in I_m}in N^m be a finite sequence (of dimensions) and a=(a_i)_{i in I_n} be a sequence of positive numbers (of weights), where I_k={1,... ,k} for k in N. We introducethe (a, d)-designs i.e., m-tuples Phi=(F_j)_{j in I_m} such that F_j={f_{ij}}_{i in I_n} is a finite sequence in C^{d_j}, j in I_m, and such that the sequence of non-negative numbers (|f_{ij}|^2)_{j in I_m} forms a partition of a_i, i in I_n. We characterize the existence of (a, d)-designs with prescribed properties in terms of majorization relations. We show, by means of a finite-step algorithm, that there exist (a, d)-designs Phi^{op}=( F_j^{ op})_{j in I_m} that are universally optimal; that is, for every convex function varphi:[0,infty) to [0,infty) then Phi^{op} minimizes the joint convex potential induced by varphi among (a, d)-designs, namely sum_{j in I_m} P_varphi(F_j^{ op})< sum_{j in I_m}P_varphi(F_j) for every (a, d)-design Phi=( F_j)_{j in I_m}, where P_varphi(F)=tr(varphi(S_F)); in particular, Phi^{op} minimizes both the joint frame potential and the joint mean square error among (a, d)-designs. We show that in this case F_j^{op} is a frame for C^{d_j}, for j in I_m. This corresponds to the existence of optimal encoding-decoding schemes for multitasking devices with energy restrictions.