Linear Universal Decoding for Compound Channels: a Local to Global Geometric Approach

TL;DR

This paper introduces a local geometric approach to construct linear decoders that achieve capacity in compound discrete memoryless channels, addressing a gap in existing decoder designs and enabling capacity-achieving solutions.

Contribution

It develops a novel local geometric analysis method that allows constructing linear decoders for compound channels, which was previously unresolved.

Findings

Existence of linear decoders achieving capacity for compound DMCs.

Development of a local geometric analysis technique.

Reduction of the problem to an inner product space in noisy channels.

Abstract

Over discrete memoryless channels (DMC), linear decoders (maximizing additive metrics) afford several nice properties. In particular, if suitable encoders are employed, the use of decoding algorithm with manageable complexities is permitted. Maximum likelihood is an example of linear decoder. For a compound DMC, decoders that perform well without the channel's knowledge are required in order to achieve capacity. Several such decoders have been studied in the literature. However, there is no such known decoder which is linear. Hence, the problem of finding linear decoders achieving capacity for compound DMC is addressed, and it is shown that under minor concessions, such decoders exist and can be constructed. This paper also develops a "local geometric analysis", which allows in particular, to solve the above problem. By considering very noisy channels, the original problem is reduced, in the limit, to an inner product space problem, for which insightful solutions can be found. The local setting can then provide counterexamples to disproof claims, but also, it is shown how in this problem, results proven locally can be "lifted" to results proven globally.