Optimal frame designs for multitasking devices with weight restrictions

Abstract

Let $\mathbf{d}=(d_j{)}_{j\in {\mathbb{I}}_m}\in {\mathbb{N}}^m$ be a finite sequence (of dimensions) and $\alpha =({\alpha }_i{)}_{i\in {\mathbb{I}}_n}$ be a sequence of positive numbers (of weights), where ${\mathbb{I}}_k=\{1,\dots ,k\}$ for $k\in \mathbb{N}$. We introduce the $(\alpha ,\mathbf{d})$-designs i.e., $m$-tuples $\Phi =({\mathcal{F}}_j{)}_{j\in {\mathbb{I}}_m}$ such that ${\mathcal{F}}_j=\{f_{ij}{\}}_{i\in {\mathbb{I}}_n}$ is a finite sequence in ${\mathbb{C}}^{d_j}$, $j\in {\mathbb{I}}_m$, and such that the sequence of non-negative numbers $(\Vert f_{ij}{\Vert }^2{)}_{j\in {\mathbb{I}}_m}$ forms a partition of ${\alpha }_i$, $i\in {\mathbb{I}}_n$. 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 ${\Phi }^{\mathrm{op}}=({\mathcal{F}}_j^{\mathrm{op}}{)}_{j\in {\mathbb{I}}_m}$ that are universally optimal; that is, for every convex function $\phi :[0,\infty )\to [0,\infty )$ then ${\Phi }^{\mathrm{op}}$ minimizes the joint convex potential induced by $\phi$ among $(\alpha ,\mathbf{d})$-designs, namely $$\sum _{j\in {\mathbb{I}}_m}{\text{P}}_{\phi }({\mathcal{F}}_j^{\mathrm{op}})\le \sum _{j\in {\mathbb{I}}_m}{\text{P}}_{\phi }({\mathcal{F}}_j)$$ for every $(\alpha ,\mathbf{d})$-design $\Phi =({\mathcal{F}}_j{)}_{j\in {\mathbb{I}}_m}$, where ${\text{P}}_{\phi }(\mathcal{F})=tr(\phi (S_{\mathcal{F}}))$; in particular, ${\Phi }^{\mathrm{op}}$ minimizes both the joint frame potential and the joint mean square error among $(\alpha ,\mathbf{d})$-designs. We show that in this case ${\mathcal{F}}_j^{\mathrm{op}}$ is a frame for ${\mathbb{C}}^{d_j}$, for $j\in {\mathbb{I}}_m$. This corresponds to the existence of optimal encoding-decoding schemes for multitasking devices with energy restrictions.

19 pages. This is version of the paper which was published, with mayor changes (one of them in the title) induced by a new approach which reduced the lenght (the original had 34 pages)

References in corpus (1)

• Frame completions for optimally robust reconstruction