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Multi-Sensor Distributed Hypothesis Testing in the Low-Power Regime

Cécile Bouette, Michèle Wigger

TL;DR

This work investigates distributed binary hypothesis testing with two sensors communicating over a memoryless MAC under sublinear input-cost budgets, focusing on Stein's error exponent. It shows that for a broad class of channels, including generalized Gaussian MACs and fully-connected DMMACs, sensor communication does not improve the Stein-exponent, which collapses to the local exponent $D(P_V||Q_V)$. The results extend prior zero-rate findings to multi-sensor and continuous-channel settings, and they provide exact exponent characterizations for DMMACs with arbitrary costs based on the channel's connectivity class (full, sparse, etc.). Achievability and converse proofs demonstrate when zero-rate noiseless-link reductions are tight, and when the exponent can still match a zero-rate configuration for partially-connected channels. Collectively, the paper clarifies how stringent resource limits at sensors influence the value of distributed sensing in hypothesis testing and guides the design of low-power IoT sensing networks.

Abstract

We characterize the Stein-exponent of a distributed hypothesis testing scenario where two sensors transmit information through a memoryless multiple access channel (MAC) subject to a sublinear input cost constraint with respect to the number of channel uses and where the decision center has access to an additional local observation. Our main theorem provides conditions on the channel and cost functions for which the Stein-exponent of this distributed setup is no larger than the Stein-exponent of the local test at the decision center. Under these conditions, communication from the sensors to the decision center is thus useless in terms of Stein-exponent. The conditions are satisfied for additive noise MACs with generalized Gaussian noise under a p-th moment constraint (including the Gaussian channel with second-moment constraint) and for the class of fully-connected (where all inputs can induce all outputs) discrete memoryless multiple-access channels (DMMACs) under arbitrary cost constraints. We further show that for DMMACs that are not fully-connected, the Stein-exponent is larger and coincides with that of a setup with zero-rate noiseless communication links from either both sensors or only one sensor, as studied in [1].

Multi-Sensor Distributed Hypothesis Testing in the Low-Power Regime

TL;DR

This work investigates distributed binary hypothesis testing with two sensors communicating over a memoryless MAC under sublinear input-cost budgets, focusing on Stein's error exponent. It shows that for a broad class of channels, including generalized Gaussian MACs and fully-connected DMMACs, sensor communication does not improve the Stein-exponent, which collapses to the local exponent . The results extend prior zero-rate findings to multi-sensor and continuous-channel settings, and they provide exact exponent characterizations for DMMACs with arbitrary costs based on the channel's connectivity class (full, sparse, etc.). Achievability and converse proofs demonstrate when zero-rate noiseless-link reductions are tight, and when the exponent can still match a zero-rate configuration for partially-connected channels. Collectively, the paper clarifies how stringent resource limits at sensors influence the value of distributed sensing in hypothesis testing and guides the design of low-power IoT sensing networks.

Abstract

We characterize the Stein-exponent of a distributed hypothesis testing scenario where two sensors transmit information through a memoryless multiple access channel (MAC) subject to a sublinear input cost constraint with respect to the number of channel uses and where the decision center has access to an additional local observation. Our main theorem provides conditions on the channel and cost functions for which the Stein-exponent of this distributed setup is no larger than the Stein-exponent of the local test at the decision center. Under these conditions, communication from the sensors to the decision center is thus useless in terms of Stein-exponent. The conditions are satisfied for additive noise MACs with generalized Gaussian noise under a p-th moment constraint (including the Gaussian channel with second-moment constraint) and for the class of fully-connected (where all inputs can induce all outputs) discrete memoryless multiple-access channels (DMMACs) under arbitrary cost constraints. We further show that for DMMACs that are not fully-connected, the Stein-exponent is larger and coincides with that of a setup with zero-rate noiseless communication links from either both sensors or only one sensor, as studied in [1].

Paper Structure

This paper contains 17 sections, 4 theorems, 30 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Notation
  3. General Model
  4. Results
  5. Generalized Gaussian channels with a $p$-th moment constraint
  6. Discrete memoryless channels with arbitrary cost constraints
  7. Proof of Theorem \ref{['thm1']}
  8. Proofs of Theorem \ref{['cor2']}
  9. Proof of Theorem \ref{['theorem_dmc']}
  10. Proof of \ref{['it:th1']})
  11. Proof of \ref{['it:th2']})
  12. Proof of \ref{['it:th3']})
  13. Achievability
  14. Converse
  15. Proof of \ref{['it:th4']})
  16. ...and 2 more sections

Key Result

Proposition 1

Allowing the decision center to take a randomized decision does not increase the Stein-exponent $\theta^*_{\textnormal{sublin}}$.

Figures (4)

  • Figure 1: Distributed hypothesis testing over a discrete memoryless channel.
  • Figure 2: Randomized local hypothesis test.
  • Figure 3: Enhanced distributed hypothesis testing setup where the output consists of the pair $(Y,X_1)$.
  • Figure 4: Randomized hypothesis test.

Theorems & Definitions (10)

  • Definition 1
  • Proposition 1
  • Theorem 1
  • Remark 1
  • Remark 2
  • Theorem 2
  • Example 1
  • Theorem 3
  • Remark 3
  • Remark 4