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 $\mathbb {I}_{k}=\{1,\ldots ,k\}$ for $k\in \mathbb {N}$. We introduce the (α, 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 $(\|f_{ij}\|^{2})_{j\in \mathbb {I}_{m}}$ forms a partition of α_i, $i\in \mathbb {I}_{n}$. 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 ${\Phi }^{\text {op}}=(\mathcal {F}_{j}^{\text {op}})_{j\in \mathbb {I}_{m}}$ that are universally optimal; that is, for every convex function $\varphi :[0,\infty )\rightarrow [0,\infty )$, then Φ^op 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 ${\Phi }=(\mathcal F_{j})_{j\in \mathbb {I}_{m}}$, where $\text {P}_{\varphi }(\mathcal F)=\text {tr}(\varphi (S_{\mathcal {F}}))$; 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, $\mathcal {F}_{j}^{\text {op}}$ 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.