Extended Functional Representation Lemma: A Tool For Privacy, Semantic Representation, Caching, and Compression Design

TL;DR

This work surveys an information-theoretic framework for privacy mechanism design using Extended Functional Representation Lemma variants to achieve privacy-utility trade-offs under non-zero leakage, per-letter privacy constraints, and prioritized private data. It develops bounds and design principles (via $\text{EFRL}$/$\text{ESFRL}$) for both hidden and observable private-data scenarios and shows how these tools yield low-complexity, near-optimal mappings. The methodology is instantiated across applications including semantic communications, caching/delivery, and compression, with extensions to multi-user and fair representations. The results highlight the practical relevance of constructive privacy mappings and suggest promising directions in information geometry and alternative leakage measures for future research.

Abstract

This paper provides an overview of a problem in information-theoretic privacy mechanism design, addressing two scenarios in which private data is either observable or hidden. In each scenario, different privacy measures are used, including bounded mutual information and two types of per-letter privacy constraints. Considering the first scenario, an agent observes useful data that is correlated with private data, and wants to disclose the useful information to a user. Due to the privacy concerns, direct disclosure is prohibited. Hence, a privacy mechanism is designed to generate disclosed data which maximizes the revealed information about the useful data while satisfying a privacy constraint. In the second scenario, the agent has additionally access to the private data. We discuss how the Functional Representation Lemma, the Strong Functional Representation Lemma, and their extended versions are useful for designing low-complexity privacy mechanisms that achieve optimal privacy-utility trade-offs under certain constraints. Furthermore, another privacy design problem is presented where part of the private attribute is more private than the remaining part. Finally, we provide applications including semantic communications, caching and delivery, and compression designs, where the approach can be applied.

Figures (10)

• Figure 1: In the first scenario the agent has only access to $Y$ and in the second scenario the agent has access to both $X$ and $Y$.
• Figure 2: Comparing the upper bounds obtained in kostala and shah for $BSC(\theta)$. The blue curve illustrates the upper bound found in kostala and the red line shows the upper bound found in shah.
• Figure 3: Possible positions of the optimizers. $\mathbb{S}_u^*(i)$ is inside an $\ell_1$-ball of radius $r$ with center $\mathbb{S}^*(i)$.
• Figure 4: For the privacy mechanism design, we are looking for $L^*$ in the red region (vector space A) which results in a vector with the largest Euclidean norm in vector space D. Space B and space C are probability spaces for the input and output distributions, the circle in space A represents the vectors that satisfy the strong $\chi^2$-privacy criterion and the red region denotes all vectors that are orthogonal to vector $\sqrt{P_X}$. Starting from Space A and reaching Space D the mapping between Space A and Space D can be found as $W=[\sqrt{P_Y}^{-1}]P_{X|Y}^{-1}[\sqrt{P_X}]$.
• Figure 5: A server wants to send a response over a shared link to satisfy users$'$ demands, but since the database is correlated with the private data existing schemes are not applicable. In the delivery phase, we hide the information about $X$ using one-time-pad coding and send the rest of response using Functional Representation Lemma (FRL).

Theorems & Definitions (1)

• Example 1