- The paper introduces posterior matching, a novel principle for feedback communication over memoryless channels that achieves rates near mutual information while simplifying transmission.
- This framework provides a unified understanding for existing feedback schemes like Horstein for BSC and Schalkwijk-Kailath for AWGN channels, demonstrating its broad applicability.
- Under regularity conditions, the posterior matching scheme is proven capacity-achieving, with error probabilities tied to the convergence rates of associated iterated function systems.
Optimal Feedback Communication via Posterior Matching
The paper "Optimal Feedback Communication via Posterior Matching" by Ofer Shayevitz and Meir Feder explores a new principle for communication over memoryless channels with noiseless feedback, termed "posterior matching." The authors propose a transmission scheme based on this principle that seamlessly integrates feedback into the communication process, simplifying the transmission strategy while achieving rates close to mutual information. This paradigm provides a unified approach that encompasses well-known schemes, such as the Horstein scheme for Binary Symmetric Channels (BSC) and the Schalkwijk-Kailath scheme for Additive White Gaussian Noise (AWGN) channels.
Overview of the Posterior Matching Principle
Feedback in memoryless channels traditionally does not increase capacity, yet it can greatly simplify transmission schemes to achieve capacity. Posterior matching utilizes the feedback to align the posterior distribution of the message with the input distribution required by the channel. In essence, the transmitted signal should represent the information missing at the receiver in an optimal way based on the observed outputs. This framework allows for a recursive representation of transmission functions, making the system run effectively "on the fly" with minimal computational complexity.
Recursive Scheme Representation
The authors present a recursive characterization of the posterior matching scheme using the inverse c.d.f. of the input and the inverse channel's distribution function. The recursive formulation is particularly useful for continuous distributions, where it represents a simple yet effective modeling of communication. It shows that the feedback mechanism can emulate the desired input/output distributions, offering a simplified yet robust transmission strategy that is capacity achieving under general conditions.
Application to Known Feedback Schemes
The paper demonstrates that classical feedback schemes fit within the posterior matching framework. Specifically, it shows that the Horstein scheme for BSC can be interpreted as slicing the posterior distribution at its median, reflecting a binary representation that achieves the required input distribution. Similarly, the Schalkwijk-Kailath scheme for AWGN channels is rephrased as transmitting the MMSE error term with appropriate scaling to maintain the power constraint, highlighting the innate simplicity and effectiveness of such an approach when applied universally.
Achievability and Error Analysis
A core result presented is that under certain regularity conditions, the posterior matching scheme achieves any rate below the mutual information for a wide class of input/channel pairs, whether discrete or continuous. The essence lies in proving that the posterior pdf around the message concentrates sufficiently as transmission progresses, allowing for accurate decoding even at high rates.
Additionally, the authors delve into error probability analysis, demonstrating that convergence rates of iterated function systems (IFS) dictate achievable rates and error probabilities. Such an approach provides explicit insights into the trade-offs involved in feedback communication, offering guidelines on designing systems with lower error performance.
Extensions and Implications
The paper discusses the flexibility of the posterior matching scheme through variants and extensions, such as the inclusion of a different ordering of inputs to handle fixed-point phenomena in certain channels. Furthermore, considerations on channel model mismatch indicate that the scheme maintains some robustness against unanticipated channel changes.
Conclusion and Future Work
This work provides a robust yet straightforward framework for feedback-based communication that neatly ties together various classical schemes while inviting potential future research avenues. Specifically, its application to channels with memory and possible universal adaptations across different channel models are promising areas for further exploration. The innovative use of posterior matching could lead to more effective communication systems exploiting feedback capabilities, enhancing reliability for numerous practical scenarios.