[论文解读] Topology of Privacy: Lattice Structures and Information Bubbles for Inference and Obfuscation

Michael Erdmann|arXiv (Cornell University)|Dec 12, 2017

Privacy-Preserving Technologies in Data参考文献 20被引用 6

一句话总结

本文提出了一种基于格结构和由个体与属性间关系导出的单纯复形的拓扑框架，用于建模隐私。通过应用Dowker定理和同调理论，将隐私损失表征为拓扑坍塌，并识别出各向同性和球形孔为保护属性和关联隐私的关键不变量，该方法在动态推理和混淆策略中具有应用价值。

ABSTRACT

Information has intrinsic geometric and topological structure, arising from relative relationships beyond absolute values or types. For instance, the fact that two people share a meal describes a relationship independent of the meal's ingredients. Multiple such relationships give rise to relations and their lattices. Lattices have topology. That topology informs the ways in which information may be observed, hidden, inferred, and dissembled. Dowker's Theorem establishes a homotopy equivalence between two simplicial complexes derived from a relation. From a privacy perspective, one complex describes individuals with common attributes, the other describes attributes shared by individuals. The homotopy equivalence produces a lattice. An element in the lattice consists of two components, one being a set of individuals, the other being a set of attributes. The lattice operations join and meet each amount to set intersection in one component and set union followed by a potentially privacy-puncturing inference in the other component. Privacy loss appears as simplicial collapse of free faces. Such collapse is local, but the property of fully preserving both attribute and association privacy requires a global condition: a particular kind of spherical hole. By looking at the link of an identifiable individual in its encompassing Dowker complex, one can characterize that individual's attribute privacy via another sphere condition. Even when long-term attribute privacy is impossible, homology provides lower bounds on how an individual may defer identification, when that individual has control over how to reveal attributes. Intuitively, the idea is to first reveal information that could otherwise be inferred. This last result highlights privacy as a dynamic process. Privacy loss may be cast as gradient flow. Harmonic flow for privacy preservation may be fertile ground for future research.

研究动机与目标

• 通过格和单纯复形结构，将隐私形式化为关系数据的拓扑属性。
• 理解隐私损失如何在Dowker复形中表现为拓扑坍塌。
• 通过同调条件（特别是球形孔）表征个体和群体的属性隐私。
• 开发基于格的推理控制策略，实现动态属性释放以延迟识别。
• 在组合隐私框架内建模随机感知和传感器噪声。

提出的方法

• 从二元关系R构建两个对偶的单纯复形：一个基于共享属性的个体，另一个基于共享个体的属性。
• 应用Dowker定理，在两个复形之间建立同伦等价，揭示具有格结构的共同核心。
• 将隐私损失建模为单纯复形中自由面的坍塌，完全隐私保护要求全局球形孔条件。
• 通过Dowker复形中个体的邻域定义属性隐私，采用球面条件，并推广至群体。
• 将各向同性定义为一种拓扑条件，即格结构呈现均匀性，表明其随时间可收缩，从而造成隐私破裂。
• 引入按偏序链排序的属性释放序列以延迟识别，同调提供隐私持续时间的下界。

实验结果

研究问题

• RQ1 如何利用格结构和单纯复形对关系数据中的隐私拓扑进行建模？
• RQ2 拓扑坍塌（例如自由面删除）在何种方式下对应于推理攻击中的隐私损失？
• RQ3 哪些拓扑条件——特别是球形孔——可确保属性和关联隐私的保护？
• RQ4 如何利用属性披露的顺序来延迟识别？同调在量化隐私持续时间中起什么作用？
• RQ5 随机传感器观测如何映射到组合隐私框架中？什么是重心剖分在建模不确定性中的作用？

主要发现

• 隐私损失在拓扑上被表征为Dowker复形中自由面的坍塌，完全隐私保护要求全局球形孔条件。
• 若个体在Dowker复形中的邻域同伦等价于球面，则个体属性隐私得以保护，通过高维球面条件可推广至群体。
• Dowker复形的同调为识别个体所需步数提供了下界，即使长期隐私无法实现亦成立。
• 由于推理导致的单纯复形坍塌是局部的，但保护属性和关联隐私需要全局拓扑条件——具体而言，即存在球形孔。
• 个体与属性之间Galois连接的格结构支持动态混淆策略，包括可提供信息的释放序列，以延迟识别。
• 随机传感器观测可通过属性单形的重心剖分进行建模，犹豫区域映射到空单形，所得决策空间同构于单形边界的第一级重心剖分。