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Decentralised Variational Inference Frameworks for Multi-object Tracking on Sensor Networks: Additional Notes

Qing Li, Runze Gan, Simon Godsill

TL;DR

A decentralised gradient-based VI framework that optimises the Locally Maximised Evidence Lower Bound (LM-ELBO) instead of the standard ELBO, which reduces the parameter search space and enables faster convergence, making it particularly beneficial for decentralised tracking.

Abstract

This paper tackles the challenge of multi-sensor multi-object tracking by proposing various decentralised Variational Inference (VI) schemes that match the tracking performance of centralised sensor fusion with only local message exchanges among neighboring sensors. We first establish a centralised VI sensor fusion scheme as a benchmark and analyse the limitations of its decentralised counterpart, which requires sensors to await consensus at each VI iteration. Therefore, we propose a decentralised gradient-based VI framework that optimises the Locally Maximised Evidence Lower Bound (LM-ELBO) instead of the standard ELBO, which reduces the parameter search space and enables faster convergence, making it particularly beneficial for decentralised tracking. This proposed framework is inherently self-evolving, improving with advancements in decentralised optimisation techniques for convergence guarantees and efficiency. Further, we enhance the convergence speed of proposed decentralised schemes using natural gradients and gradient tracking strategies. Results verify that our decentralised VI schemes are empirically equivalent to centralised fusion in tracking performance. Notably, the decentralised natural gradient VI method is the most communication-efficient, with communication costs comparable to suboptimal decentralised strategies while delivering notably higher tracking accuracy.

Decentralised Variational Inference Frameworks for Multi-object Tracking on Sensor Networks: Additional Notes

TL;DR

A decentralised gradient-based VI framework that optimises the Locally Maximised Evidence Lower Bound (LM-ELBO) instead of the standard ELBO, which reduces the parameter search space and enables faster convergence, making it particularly beneficial for decentralised tracking.

Abstract

This paper tackles the challenge of multi-sensor multi-object tracking by proposing various decentralised Variational Inference (VI) schemes that match the tracking performance of centralised sensor fusion with only local message exchanges among neighboring sensors. We first establish a centralised VI sensor fusion scheme as a benchmark and analyse the limitations of its decentralised counterpart, which requires sensors to await consensus at each VI iteration. Therefore, we propose a decentralised gradient-based VI framework that optimises the Locally Maximised Evidence Lower Bound (LM-ELBO) instead of the standard ELBO, which reduces the parameter search space and enables faster convergence, making it particularly beneficial for decentralised tracking. This proposed framework is inherently self-evolving, improving with advancements in decentralised optimisation techniques for convergence guarantees and efficiency. Further, we enhance the convergence speed of proposed decentralised schemes using natural gradients and gradient tracking strategies. Results verify that our decentralised VI schemes are empirically equivalent to centralised fusion in tracking performance. Notably, the decentralised natural gradient VI method is the most communication-efficient, with communication costs comparable to suboptimal decentralised strategies while delivering notably higher tracking accuracy.
Paper Structure (70 sections, 1 theorem, 123 equations, 7 figures, 3 tables, 4 algorithms)

This paper contains 70 sections, 1 theorem, 123 equations, 7 figures, 3 tables, 4 algorithms.

Table of Contents

  1. Introduction
  2. Contribution
  3. Paper Outline
  4. Problem Formulation and Modelling
  5. Dynamical Model
  6. NHPP Measurement Model and Association Prior
  7. Variational Filtering for Multi-object Tracking
  8. Prediction step in the variational filtering
  9. Update step in the variational filtering
  10. Variational Inference with Standard ELBO for Multi-object Tracking
  11. Variational Inference for General Problem Settings
  12. Variational Inference Update for Multi-object Tracking
  13. Standard free-form ELBO and centralised CAVI for sensor fusion and tracking
  14. Decentralised consensus-based CAVI
  15. Standard fixed-form ELBO and gradient-based variational inference for sensor fusion and tracking
  16. ...and 55 more sections

Key Result

Lemma C.1

Recall the assumption from Section sec: LM-ELBO over O-ELBO, where $q_n(\theta_n; \rho_n) = \prod_{s=1}^{N_s} q_n(\theta_n^s; \rho_n^s)$ and $\rho_n = [\rho_n^1, \rho_n^2, \dots, \rho_n^{N_s}]$. Let $\rho_n^{*}(\lambda_n)$ be the parameter value of $q_n(\theta_n;\rho_n)$ that yields the optimal dist then we have where $\rho_n^{s*}(\lambda_n)$$(s=1,2,...,N_s)$ is defined in Section sec: local LM-E

Figures (7)

  • Figure 1: Sensor networks of dataset 1 and 2 in Scene 1; Red circles are sensor nodes, grey lines denote their connectivity, and black dots are an example measurement data of one time step at a single sensor
  • Figure 2: Example tracking performance at one Monte Carlo run of DeNG-VT-GT (left) and DeAA-VT (right); coloured dotted lines are estimate, black lines are ground truth and grey ellipses are 95% confidence interval. The boxes in the right figure mark the track loss events using DeAA-VT
  • Figure 3: GOSPA over iteration number at a single time step; lines and shaded area are mean and $\pm1$ standard deviation of GOSPA value averaged over all sensors, respectively.
  • Figure 4: GOSPA over 50 time steps; for all methods, lines are means of GOSPA averaged over all sensors and shaded areas indicate $\pm1$ standard deviation. Y-axis is log-scale.
  • Figure 5: Time varying sensor networks; Red circles are sensor nodes and grey lines indicate their connectivity. Black dots are measurements received at 15th time step from 1st sensor (left) and 38th time step from 10th sensor (right)
  • ...and 2 more figures

Theorems & Definitions (2)

  • Lemma C.1
  • proof