Building on the well-established capacity-achieving schemes of Sun-Jafar (for replicated storage) and the closely related scheme of Banawan-Ulukus (for MDS-coded setting), a recent work by Chandan et al. proposed new classes of weak private information retrieval (WPIR) schemes for the collusion-free (replication and MDS-coded) setting, as well as for the $T$-colluding scenario. In their work, Chandan et al. characterized the expressions for the rate-privacy trade-offs for these classes of WPIR schemes, under the mutual information leakage and maximal leakage metrics. Explicit achievable trade-offs for the same were also presented, which were shown to be competitive or better than prior WPIR schemes. However, the class-wise optimality of the reported trade-offs were unknown. In this work, we show that the explicit rate-privacy trade-offs reported for the Sun-Jafar-type schemes by Chandan et al. are optimal for the non-colluding and replicated setting. Furthermore, we prove the class-wise optimality for Banawan-Ulukus-type MDS-WPIR and Sun-Jafar-type $T$-colluding WPIR schemes, under threshold-constraints on the system parameters. When these threshold-constraints do not hold, we present counter-examples which show that even higher rates than those reported before can be achieved.

翻译：基于Sun-Jafar（针对复制存储）的成熟容量可达方案以及密切相关的Banawan-Ulukus方案（针对MDS编码设置），Chandan等人的近期工作提出了针对无合谋（复制与MDS编码）场景以及$T$-合谋场景的新型弱私有信息检索（WPIR）方案类别。在他们的工作中，Chandan等人刻画了这些WPIR方案类别在互信息泄漏和最大泄漏度量下的速率-隐私权衡表达式，并给出了相应的显式可达权衡，其性能被证明优于或可与先前的WPIR方案相媲美。然而，所报告权衡的类内最优性尚属未知。本工作中，我们证明Chandan等人针对Sun-Jafar型方案报告的显式速率-隐私权衡在无合谋复制存储场景下是最优的。此外，我们在系统参数的阈值约束条件下，证明了Banawan-Ulukus型MDS-WPIR方案与Sun-Jafar型$T$-合谋WPIR方案的类内最优性。当这些阈值约束不成立时，我们通过反例表明，可以实现比先前报告更高的速率。