Optimal frame designs for multitasking devices with weight restrictions

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

ADVANCES IN COMPUTATIONAL MATHEMATICS

SPRINGER

Lugar: Berlin; Año: 2020 vol. 46

1019-7168

Let $mathbf d=(d_j)_{jinmathbb I_m}inmathbb N^m$ be a finite sequence (of dimensions) and $alpha=(alpha_i)_{iinmathbb I_n}$ be a sequence of positive numbers (of weights), where $mathbb I_k={1,ldots,k}$ for $kinmathbb N$. We introduce the $(alpha, , ,mathbf d)$-designs i.e., $m$-tuples $Phi=(mathcal F_j)_{jinmathbb I_m}$ such that $mathcal F_j= f_{ij}}_{iinmathbb I_n}$ is a finite sequence in $mathbb C^{d_j}$, $jinmathbb I_m$, and such that the sequence of non-negative numbers $(|f_{ij}|^2)_{jinmathbb I_m}$ forms a partition of $alpha_i$, $iinmathbb I_n$. We characterize the existence of $(alpha, , , mathbf d)$-designs with prescribed properties in terms of majorization relations. We show, by means of a finite-step algorithm, that there exist $(alpha, , , mathbf d)$-designs $Phi^{m op}=(mathcal F_j^{m op})_{jinmathbb I_m}$ that are universally optimal; that is, for every convex function $arphi:[0,infty)ightarrow [0,infty)$ then $Phi^{m op}$ minimizes the joint convex potential induced by $arphi$ among $(alpha, , , mathbf d)$-designs, namely $$ sum_{jinmathbb I_m}ext{P}_arphi(mathcal F_j^{m op})leq sum_{jin mathbb I_m}ext{P}_arphi(mathcal F_j) $$ for every $(alpha, , , mathbf d)$-design $Phi=(mathcal F_j)_{jinmathbb I_m}$, where $ext{P}_arphi(cF)=r(arphi(S_cF))$; in particular, $Phi^{m op}$ minimizes both the joint frame potential and the joint mean square error among $(alpha, , , mathbf d)$-designs. We show that in this case $cF_j^{m op}$ is a frame for $C^{d_j}$, for $jinI_m$. This corresponds to the existence of optimal encoding-decoding schemes for multitasking devices with energy restrictions.