Efficient Function Computation and Identification over Locally Homomorphic Multiple-Access Channels

Core Concepts

Locally homomorphic channels provide an efficient way to compute functions and perform identification over multiple-access channels, enabling significant rate improvements compared to naive code constructions.

Abstract

The paper develops the notion of a locally homomorphic channel (LHC) and proves an approximate equivalence between LHCs and codes for computing functions. It derives decomposition properties of LHCs, which are then used to analyze and construct codes where two messages must be encoded independently. This leads to new results for identification (ID) and K-ID, which are illustrated using the example of deterministic ID over binary symmetric channels. Key highlights: LHCs are defined as channels that preserve the structure of functions, represented as hypergraphs, with high probability. It is shown that for reliable function computation, the channel itself must be locally homomorphic, and stochastic coding provides limited gains. When messages are distributed among parties, stochastic encoding can help, and bipartite encoders are constructed from encoders that assume the receiver knows one of the messages. For the ID setting, it is demonstrated that the two encoders can be constructed independently, even when both messages are sent over a multiple-access channel. The results are illustrated using the example of deterministic ID over binary symmetric channels, where significant rate improvements are achieved compared to naive code constructions. The paper discusses the challenges in optimizing the input hypergraphs to LHCs to fully explain the tradeoff between the rates of the two messages.

Stats

Pr{𝑓(𝑎) = Dec ∘𝜑∘Enc (𝑎)} ≥1 −𝜆𝑓(𝑎) Pr{𝜑(𝑎) ∈𝑓ℰ(𝐺𝑎)} ≥1 −𝜆𝐺𝑎 Pr{𝑑𝐻(𝑌𝑛, ¯ 𝑌𝑛) ≥𝑛𝜃𝛿−𝑛𝜖𝜃𝛿} ≥1 −2𝑒−𝑛𝜖2𝜃𝛿/2

Quotes

"For any function 𝑓: 𝒜→ℬand channel 𝜑: 𝒳→V(𝒴), if there exists an (𝑓, 𝜑, 𝜆)-code and 4𝜆≤𝜅≤1/2, then there exist partition hypergraphs 𝐺and 𝐹, |ℰ(𝐺)| = |ℰ(𝐹)| = |𝑓(𝒜)|, and 𝜑: 𝐺 𝜅 − →𝐹is an edge-bijective LHC." "This shows that knowing 𝐹, 𝐺1, and 𝐺2, one can design both encoders as if the other message were known to the receiver. The challenge is, however, finding a suitable 𝐹."

Key Insights Distilled From

Function Computation and Identification over Locally Homomorphic Multiple-Access Channels

by Johannes Ros... at arxiv.org 04-23-2024

https://arxiv.org/pdf/2404.14390.pdf

Deeper Inquiries

How can the input hypergraphs to LHCs be optimized to fully explain the tradeoff between the rates of the two messages in bipartite encoding?

To optimize the input hypergraphs to Locally Homomorphic Channels (LHCs) for a comprehensive explanation of the tradeoff between the rates of the two messages in bipartite encoding, several key steps can be taken: Vertex Set Structure: Ensuring that the vertex sets of the hypergraphs are well-defined and structured in a way that facilitates clear mapping and analysis. Rectangular vertex sets can be particularly useful in bipartite encoding scenarios. Edge Mapping: Developing precise and effective edge maps that accurately represent the relationships between the vertices in the hypergraphs. Edge-bijective mappings are crucial for maintaining the integrity of the information flow. Homomorphisms: Utilizing homomorphisms effectively to preserve the structure of the hypergraphs during encoding and decoding processes. This ensures that the information is transmitted accurately and efficiently. Tradeoff Analysis: Conducting thorough tradeoff analyses between the rates of the two messages to understand the impact of different encoding strategies on the overall communication efficiency. Optimization Algorithms: Implementing optimization algorithms that can iteratively refine the hypergraphs based on performance metrics and objectives related to the tradeoff between message rates. By focusing on these aspects and continuously refining the input hypergraphs based on the specific requirements of the bipartite encoding scenario, a more detailed and insightful explanation of the rate tradeoff can be achieved.

How can the input hypergraphs to LHCs be optimized to fully explain the tradeoff between the rates of the two messages in bipartite encoding?

The fundamental limits on the achievable rates for function computation and identification over multiple-access channels when both messages are unknown to the receiver are influenced by several factors: Channel Capacity: The inherent capacity of the multiple-access channel plays a significant role in determining the maximum achievable rates for function computation and identification. Shannon's capacity theorem provides insights into these limits. Noise and Interference: The presence of noise and interference in the channel can constrain the achievable rates, especially when both messages are unknown to the receiver. Mitigating noise and interference is crucial for improving communication efficiency. Encoding and Decoding Strategies: The effectiveness of encoding and decoding strategies, particularly in scenarios where both messages are unknown, directly impacts the achievable rates. Optimizing these strategies can help push the limits of communication rates. Information Theory Principles: Leveraging principles from information theory, such as entropy, mutual information, and coding theory, can provide theoretical bounds on the achievable rates for function computation and identification over multiple-access channels. Tradeoff Analysis: Conducting detailed tradeoff analyses between error rates, channel capacities, and encoding complexities can reveal the fundamental limits on achievable rates in such communication scenarios. By considering these factors and exploring the interplay between them, a clearer understanding of the fundamental limits on achievable rates for function computation and identification over multiple-access channels can be obtained.

Can the techniques developed in this work be extended to other communication tasks beyond function computation and identification?

Yes, the techniques developed in this work can be extended to various other communication tasks beyond function computation and identification. Some potential areas of extension include: Data Compression: The principles of LHCs and homomorphic channels can be applied to data compression tasks, where preserving the structure of data during transmission is crucial for efficient compression and decompression. Error Correction: Techniques for error correction coding and decoding can benefit from the insights gained in this work, especially in scenarios where multiple sources of data need to be corrected and reconstructed. Distributed Computing: The concept of locally homomorphic channels can be valuable in distributed computing environments where multiple nodes need to communicate and compute functions collaboratively while preserving data integrity. Secure Communication: The principles of homomorphic encryption and secure communication can be enhanced using the techniques developed in this work to ensure secure and reliable transmission of sensitive information. Network Optimization: Applying the optimization algorithms and tradeoff analyses from this work to network optimization tasks can help in improving data transmission efficiency and network performance. By adapting and extending the techniques and methodologies developed in this work to these diverse communication tasks, new insights and advancements can be achieved in various areas of communication and information theory.