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Distributed Detection under Stringent Resource Constraints

Abdelaziz Bounhar, Mireille Sarkiss, Michèle Wigger

TL;DR

This paper analyzes distributed binary hypothesis testing where a sensor communicates with a decision center over a DMC under stringent sublinear and expected-cost resource constraints. It unveils a connectivity-based dichotomy: partially-connected DMCs can achieve the zero-rate noiseless-exponent, while fully-connected DMCs generally force a local decision exponent in two regimes, with a partial gain under an expectation-based constraint (case 3). It provides explicit exponents E1, E2, E3 for the expected-cost setting and introduces new coding and converse techniques, including a Han-style zero-rate scheme adapted to noisy channels and change-of-measure arguments. The results illuminate how channel structure and cost constraints jointly shape the value of remote sensing communications for hypothesis testing, with implications for energy-constrained alert systems and sensor networks.

Abstract

This paper identifies the Stein-exponent of distributed detection when the sensor communicates to the decision center over a discrete memoryless channel (DMC) subject to one of three stringent communication constraints: 1) The number of channel uses of the DMC grows sublinearly in the number of source observations n; 2) The number of channel uses is n but a block-input cost constraint is imposed almost surely, which grows sublinearly in n; 3) The block-input constraint is imposed only on expectation. We identify a dichotomy in the Stein-exponent of all these setups depending on the structure of the DMC's transition law. Under any of these three constraints, when the DMC is partially-connected (i.e., some outputs cannot be induced by certain inputs) the Stein-exponent matches the exponent identified by Han and Kobayashi and by Shalaby and Papamarcou for the scenario where communication is of zero-rate but over a noiseless link. We prove our result by adapting Han's zero-rate coding strategy to partially-connected DMCs. In contrast, for fully-connected DMCs, in our scenarios 1) and 2) the Stein-exponent collapses to that of a local test at the decision center, rendering the remote sensor and communication useless. %To prove this result, we propose new converse proofs relying on change of measure arguments. In scenario 3), the sensor remains beneficial even for fully-connected DMCs, however also collapses compared to the case of a partially-connected DMC. Moreover, the Stein-exponent is larger when the expectation constraint is imposed only under the null hypothesis compared to when it is imposed under both hypotheses. To prove these results, we propose both new coding strategies and new converse proofs.

Distributed Detection under Stringent Resource Constraints

TL;DR

This paper analyzes distributed binary hypothesis testing where a sensor communicates with a decision center over a DMC under stringent sublinear and expected-cost resource constraints. It unveils a connectivity-based dichotomy: partially-connected DMCs can achieve the zero-rate noiseless-exponent, while fully-connected DMCs generally force a local decision exponent in two regimes, with a partial gain under an expectation-based constraint (case 3). It provides explicit exponents E1, E2, E3 for the expected-cost setting and introduces new coding and converse techniques, including a Han-style zero-rate scheme adapted to noisy channels and change-of-measure arguments. The results illuminate how channel structure and cost constraints jointly shape the value of remote sensing communications for hypothesis testing, with implications for energy-constrained alert systems and sensor networks.

Abstract

This paper identifies the Stein-exponent of distributed detection when the sensor communicates to the decision center over a discrete memoryless channel (DMC) subject to one of three stringent communication constraints: 1) The number of channel uses of the DMC grows sublinearly in the number of source observations n; 2) The number of channel uses is n but a block-input cost constraint is imposed almost surely, which grows sublinearly in n; 3) The block-input constraint is imposed only on expectation. We identify a dichotomy in the Stein-exponent of all these setups depending on the structure of the DMC's transition law. Under any of these three constraints, when the DMC is partially-connected (i.e., some outputs cannot be induced by certain inputs) the Stein-exponent matches the exponent identified by Han and Kobayashi and by Shalaby and Papamarcou for the scenario where communication is of zero-rate but over a noiseless link. We prove our result by adapting Han's zero-rate coding strategy to partially-connected DMCs. In contrast, for fully-connected DMCs, in our scenarios 1) and 2) the Stein-exponent collapses to that of a local test at the decision center, rendering the remote sensor and communication useless. %To prove this result, we propose new converse proofs relying on change of measure arguments. In scenario 3), the sensor remains beneficial even for fully-connected DMCs, however also collapses compared to the case of a partially-connected DMC. Moreover, the Stein-exponent is larger when the expectation constraint is imposed only under the null hypothesis compared to when it is imposed under both hypotheses. To prove these results, we propose both new coding strategies and new converse proofs.
Paper Structure (16 sections, 4 theorems, 27 equations, 3 figures)

This paper contains 16 sections, 4 theorems, 27 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Distributed Detection with a Sublinear Number of Channel Uses
  3. Problem Setup
  4. Results
  5. Distributed Detection with Sublinear Resources
  6. Expected Resource Constraints
  7. Summary
  8. Converse Proof of Theorem \ref{['thm_sublinear']}, Equation \ref{['eq:Ib']}
  9. Achievability Proof of Theorem \ref{['thm_sublinear']}, Equation \ref{['eq:Ia']}
  10. Converse Proof of Theorem \ref{['thm:resources']}
  11. Achievability Proof of Theorem \ref{['thm:expected']}
  12. Proof of Achievability for Equation \ref{['eq:IIIb']}
  13. Proof of Achievability for Equation \ref{['eq:IIId']}
  14. Converse Proof for Theorem \ref{['thm:expected']}
  15. Upper Bounds $E_1$ and $E_3$
  16. ...and 1 more sections

Key Result

Theorem 1

Fix $\epsilon \in [0,1)$.

Figures (3)

  • Figure 1: Distributed Hypothesis testing with a sublinear number of channel uses.
  • Figure 2: System setup with a linear number of channel uses.
  • Figure 3: Derived binary hypothesis test with channel $P_{Y^n|V^n, \mathcal{H}=0}$ used under both hypotheses.

Theorems & Definitions (8)

  • Definition 1
  • Theorem 1
  • Remark 1: Finite Values of $k$
  • Remark 2
  • Corollary 2: When communication never helps
  • Definition 2
  • Theorem 3
  • Theorem 4