弥合差分隐私与随机三角形
Bridging Differential Privacy and Random Triangles
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中文总结 AI 辅助
本研究提出两种几何表示，将差分隐私中的随机三角形映射到单纯形和半球，重构并恢复隐私损失，建立了连接差分隐私与随机形状概率分析的精确几何坐标系。
中文摘要 AI 辅助
差分隐私中高斯机制的经典分析将相邻数据集对的隐私损失简化为一个标量随机变量，该标量特征虽足以用于隐私核算，但每个扰动实例还会由敏感性向量和两个对应噪声向量形成一个高维随机三角形。本研究针对这些随机三角形提出两种互补的几何表示：第一种将归一化边长平方映射到单纯形，推导其精确联合密度、刻画其椭圆支撑，并从单纯形坐标重构经典隐私损失随机变量；第二种将每个归一化三角形的谱形状映射到半球，推导对应密度和坐标映射、恢复相同隐私损失，并刻画随维度增加的赤道漂移与带集中现象。这些结果提供了两种精确几何坐标系，补充了标量隐私损失，并将差分隐私与随机形状的概率分析相连接。
英文摘要
The classical analysis of the Gaussian mechanism in differential privacy reduces privacy loss for a pair of neighboring datasets to a scalar random variable. While this scalar characterization is sufficient for privacy accounting, each perturbation instance also induces a high-dimensional random triangle formed by the sensitivity vector and the two corresponding noise vectors. In this work, we develop two complementary geometric representations of these random triangles. The first representation maps the normalized squared edge lengths to a simplex. We derive its exact joint density, characterize its elliptical support, and reconstruct the classical privacy loss random variable from the simplex coordinates. The second representation maps the spectral shape of each normalized triangle to a hemisphere. We derive the corresponding density and coordinate mappings, recover the same privacy loss, and characterize an equatorial drift together with band concentration as the dimension increases. These results provide two exact geometric coordinate systems that complement the scalar privacy loss and connect differential privacy with the probabilistic analysis of random shapes.
