On the suboptimality of linear codes for binary distributed hypothesis testing

TL;DR

This paper investigates the limitations of linear coding schemes in binary distributed hypothesis testing, demonstrating their suboptimality compared to non-linear methods in certain scenarios.

Contribution

The paper identifies the optimality of simple truncation among linear schemes for specific tests and proves that linear codes are suboptimal for testing independence.

Findings

Truncation is optimal among linear schemes for testing opposite signs of correlation.

Linear codes are strictly suboptimal for testing independence.

Numerical evidence supports the conjecture that truncation is best for all opposite sign correlations.

Abstract

We study a binary distributed hypothesis testing problem where two agents observe correlated binary vectors and communicate compressed information at the same rate to a central decision maker. In particular, we study linear compression schemes and show that simple truncation is the best linear scheme in two cases: (1) testing opposite signs of the same magnitude of correlation, and (2) testing for or against independence. We conjecture, supported by numerical evidence, that truncation is the best linear code for testing any correlations of opposite signs. Further, for testing against independence, we also compute classical random coding exponents and show that truncation, and consequently any linear code, is strictly suboptimal.