[Paper Review] Analysis of Democratic Voting Principles used in Distributed Greedy Algorithms

Dennis Sundman, Saikat Chatterjee+1 more|arXiv (Cornell University)|Jul 16, 2014

Sparse and Compressive Sensing Techniques29 references3 citations

TL;DR

This paper provides a probabilistic analysis of democratic voting principles—majority and consensus voting—in distributed greedy pursuit algorithms for compressed sensing. By modeling local greedy algorithm performance via a single miss probability parameter, the authors prove that democratic voting improves joint support-set estimation accuracy, especially in the common and mixed support-set models, with analytical guarantees showing enhanced detection reliability under realistic error conditions.

ABSTRACT

A key aspect for any greedy pursuit algorithm used in compressed sensing is a good support-set detection method. For distributed compressed sensing, we consider a setup where many sensors measure sparse signals that are correlated via the existence of a signals' intersection support-set. This intersection support-set is called the joint support-set. Estimation of the joint support-set has a high impact on the performance of a distributed greedy pursuit algorithm. This estimation can be achieved by exchanging local support-set estimates followed by a (consensus) voting method. In this paper we endeavor for a probabilistic analysis of two democratic voting principle that we call majority and consensus voting. In our analysis, we first model the input/output relation of a greedy algorithm (executed locally in a sensor) by a single parameter known as probability of miss. Based on this model, we analyze the voting principles and prove that the democratic voting principle has a merit to detect the joint support-set.

Motivation & Objective

• To analyze the effectiveness of democratic voting principles—majority and consensus voting—in distributed greedy pursuit algorithms for compressed sensing.
• To model the input/output behavior of local greedy algorithms using a single miss probability parameter based on detection theory.
• To establish theoretical performance guarantees for joint support-set estimation in distributed compressed sensing under both common and mixed support-set models.
• To validate that democratic voting enhances detection reliability compared to individual sensor estimates, particularly in the presence of independent errors.

Proposed method

• Model the performance of local greedy pursuit algorithms using a single parameter: the probability of miss (1 - probability of detection).
• Formulate voting strategies based on consensus and majority principles to estimate the joint support-set from multiple local estimates.
• Derive analytical expressions for the conditional probability that a support-set element is correctly identified given voting outcomes across multiple sensors.
• Apply the model to both the common support-set and mixed support-set signal correlation models, using combinatorial probability and conditional independence assumptions.
• Use polynomial inequality analysis to compare the reliability of voting outcomes against individual sensor performance across varying miss probabilities.
• Verify theoretical results through numerical evaluation with fixed system parameters (N=1000, T=20, J=15, I=5), confirming the region where voting outperforms individual estimates.

Experimental results

Research questions

• RQ1 Does democratic voting—specifically majority and consensus voting—improve joint support-set estimation in distributed greedy pursuit algorithms compared to individual sensor estimates?
• RQ2 Under what conditions does consensus voting outperform individual sensor detection in terms of the probability of correctly identifying true support-set elements?
• RQ3 How does the probability of miss in local greedy algorithms affect the performance gain from democratic voting in joint support-set estimation?
• RQ4 In what parameter ranges does voting provide a guaranteed improvement in detection reliability for both common and mixed support-set models?
• RQ5 Can analytical performance guarantees be derived for democratic voting in distributed compressed sensing using a single miss probability parameter?

Key findings

• Democratic voting significantly improves the probability of correctly identifying true support-set elements compared to individual sensor estimates, especially when the local miss probability is moderate.
• For the consensus voting strategy, the probability of correct detection increases when multiple sensors agree on an element’s inclusion, with analytical bounds showing improvement in the range ε ∈ [0.0140, 0.98].
• The analysis proves that the conditional probability of correct detection given consensus across three sensors is greater than that of a single sensor, i.e., P(i ∈ T_p | i ∈ (T̂_p ∩ T̂_q ∩ T̂_r)) ≥ P(i ∈ T_p | i ∈ T̂_p), for ε ∈ [0.0140, 0.98].
• For majority voting with two out of three sensors agreeing, the improvement is guaranteed in the interval ε ∈ [0.0069, 0.98], demonstrating robustness even with higher local error rates.
• The study confirms that democratic voting principles are analytically justified and outperform individual detection in both common and mixed support-set models.
• Numerical verification using N=1000, T=20, J=15, I=5 confirms that voting consistently improves detection reliability across the relevant miss probability range.