发表机构
KTH Royal Institute of Technology; UNSW Canberra（皇家理工学院; 新南威尔士大学堪培拉校区）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对局部差分隐私离散信道，提出稀疏核机制的统一框架，精确刻画纯与近似隐私，并分解隐私缺陷为支撑泄露与重叠损失，为多种机制提供闭式公式。
AI 中文摘要
我们研究了由非负核和输入依赖的可行输出支撑集生成的稀疏局部隐私离散机制。支撑集相对于环境字母表而言刻意保持较小，并充当机制设计参数。该公式涵盖度量球支撑、稀疏指数机制、稀疏阶梯机制、稀疏随机响应、稀疏加性核（如Skellam扰动）以及泊松-二项式启发的有界支撑信道。我们给出了该类稀疏核的纯局部差分隐私和近似局部差分隐私的一般精确刻画。纯局部差分隐私迫使所有支撑集重合，因此真正输入依赖的稀疏支撑与纯隐私不相容。在近似机制中，隐私缺陷精确分解为支撑泄露和重叠超额损失。我们为几个离散机制族实例化了该公式。对于稀疏阶梯机制，壳比条件消除了重叠超额损失。对于图度量支撑，度量球的重叠对于非平凡隐私是必要的。对于稀疏随机响应，支撑泄露项以支撑失配的形式具有闭式表达式。对于一般的半径截断加性核，特别是稀疏Skellam机制，我们获得了以Skellam累积分布函数表示的精确有限支撑公式，以及精确的贝塞尔比重叠条件。数值评估说明了支撑半径和核形状如何分别影响支撑泄露和重叠超额损失。
英文摘要
We study sparse locally private discrete mechanisms generated by a nonnegative kernel and an input-dependent admissible output support. The support set is intentionally small relative to the ambient alphabet and acts as a mechanism-design parameter. This formulation covers metric-ball supports, sparse exponential mechanisms, sparse staircase mechanisms, sparse randomized response, sparse additive kernels such as Skellam perturbations, and Poisson-binomial-inspired bounded-support channels. We give a general exact characterization of pure and approximate local differential privacy for this sparse-kernel class. Pure local differential privacy forces all supports to coincide, so genuinely input-dependent sparse supports are incompatible with pure privacy. In the approximate regime, the privacy defect decomposes exactly into support leakage and overlap excess loss. We instantiate the formula for several discrete mechanism families. For sparse staircase mechanisms, a shell-ratio condition eliminates overlap excess loss. For graph-metric supports, overlap of metric balls is necessary for nontrivial privacy. For sparse randomized response, the support-leakage term has a closed form in terms of support mismatch. For generic radius-truncated additive kernels and, in particular, sparse Skellam mechanisms, we obtain exact finite-support formulas in terms of the Skellam CDF and an exact Bessel-ratio overlap condition. Numerical evaluations illustrate how support radius and kernel shape separately affect support leakage and overlap excess loss.
