On the suboptimality of linear codes for binary distributed hypothesis testing

TL;DR

The paper addresses binary distributed hypothesis testing with two rate-limited encoders observing DSBS$(p_i)$ pairs and communicates to a fusion center. It develops a Blackwell-order framework to compare linear encoders and proves that truncation is optimal among linear codes for opposite-signed correlations and independence testing, while conjecturing this extends to all opposite-signed cases. It further contrasts linear-code performance with random coding, showing that for testing against independence, linear schemes are strictly suboptimal and that a truncation-based hybrid can outperform both. These results clarify the limitations of linear coding in distributed hypothesis testing and inform practical encoder design under correlation-based hypotheses.

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.

Figures (4)

• Figure 1: Distributed hypothesis testing setup considered. The communication from $\mathsf{A}$ and $\mathsf{B}$ to $\mathsf{C}$ is constrained. We take $(X^n, Y^n) \sim \mathop{\mathrm{DSBS}}\nolimits(p_i)^{\otimes n}$ under hypothesis $\mathcal{H} = i \in \{0,1\}$ and study the performance of linear $g,h$.
• Figure 2: $(p_0, p_1)$-plane where truncation is the best linear scheme for testing $\mathop{\mathrm{DSBS}}\nolimits(p_0)$ versus $\mathop{\mathrm{DSBS}}\nolimits(p_1)$: thick, black lines where it is known (Theorem \ref{['thm: trunc']}) and red, shaded region where it is conjectured (Conjecture \ref{['conj: trunc']}).
• Figure 3: Comparison of $E_{\text{tr}}$ and $E_{\text{Han}}$ for $(p_0, p_1) = (0.1, 0.9)$. Choosing $\mathsf{P}_{U|X}$ and $\mathsf{P}_{V|Y}$ to be the same BSC seems to be the best choice of test channels in Han's scheme. At sufficiently large rates, $E_{\text{Han}}(R) > E_{\text{tr}}(R)$, and hence we have an improved exponent $E_{\text{com}}(R)$ which is strictly better than $E_{\text{tr}}(R)$ at all $R$ and strictly better than $E_{\text{Han}}(R)$ for small $R$.
• Figure 4: The hatched region contained between the two curves shows the $(p_0,p_1)$ region where $X^k \oplus Y^k \succeq_{\text{B}} G(X^n \oplus Y^n)$ for all rank-$k$, $n$-column matrices $G$, shown for three values of $(k,n) = (1,2),\,(2,3),\,(3,5)$. The region appears to be shrinking to $(p_0 - 1/2)(p_1 - 1/2) \leq 0$ as $n \to \infty$.

Theorems & Definitions (5)

• Theorem 1: Same linear code is (always) better
• proof
• Theorem 2: Truncation is (sometimes) the best linear code
• proof
• Conjecture 1