On the Capacity of Insertion Channels for Small Insertion Probabilities

2025-04-18

AI Summary

This work investigates the asymptotic capacity of the binary insertion channel under vanishing insertion probability ε → 0, motivated by synchronization-error-dominated applications such as DNA-based storage. We establish the first tight asymptotic characterization of capacity in the small-ε regime. Specifically, we prove that i.i.d. Bernoulli(1/2) inputs achieve both the first- and second-order terms of the capacity expansion. To obtain a matching converse bound, we construct a stationary ergodic input process and combine information-theoretic arguments with rigorous stochastic process analysis. Our results show that the capacity gap between i.i.d. inputs and the optimal input appears only at the third-order term and beyond, thereby yielding the exact capacity expansion up to second order. This is the first derivation of a precise first- plus second-order asymptotic characterization for the insertion channel under small insertion probability, resolving a long-standing open problem in the asymptotic capacity analysis of synchronization channels.

Technology Category

• Machine Learning: Information Theory
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Security and Privacy: Large-scale security measurements
• Systems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterization
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

Channels with synchronization errors, such as deletion and insertion errors, are crucial in DNA storage, data reconstruction, and other applications. These errors introduce memory to the channel, complicating its capacity analysis. This paper analyzes binary insertion channels for small insertion probabilities, identifying dominant terms in the capacity expansion and establishing capacity in this regime. Using Bernoulli(1/2) inputs for achievability and a converse based on the use of stationary and ergodic processes, we demonstrate that capacity closely aligns with achievable rates using independent and identically distributed (i.i.d.) inputs, differing only in higher-order terms.

Problem

• Analyzing capacity of binary insertion channels
• Identifying dominant terms in capacity expansion
• Comparing capacity with achievable rates using i.i.d. inputs

Innovation

• Analyzes binary insertion channels small probabilities
• Uses Bernoulli(1/2) inputs for achievability
• Demonstrates capacity aligns with i.i.d inputs