Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Fast computation of the TGOSPA metric for multiple target tracking via unbalanced optimal transport

Viktor Nevelius Wernholm, Alfred Wärnsäter, Axel Ringh

TL;DR

The paper tackles the computational bottleneck of evaluating the trajectory generalized optimal sub-pattern assignment (TGOSPA) metric in multi-target tracking by reformulating the LP-relaxed TGOSPA as an entropy-regularized, unbalanced multimarginal OT problem. It derives a Sinkhorn-type algorithm via block coordinate ascent in the Lagrangian dual and shows how the cost structure enables efficient projection computations with $ ext{O}(m^2 n^2)$ complexity per update, yielding linear convergence. Numerical results demonstrate substantial speed-ups over exact LP solvers while maintaining accurate approximations (≈1% relative error on larger problems), with potential for GPU acceleration. The approach provides a scalable, principled metric computation for large MTT scenarios and suggests future work on differentiable implementations for gradient-based data-driven tracking.

Abstract

In multiple target tracking, it is important to be able to evaluate the performance of different tracking algorithms. The trajectory generalized optimal sub-pattern assignment metric (TGOSPA) is a recently proposed metric for such evaluations. The TGOSPA metric is computed as the solution to an optimization problem, but for large tracking scenarios, solving this problem becomes computationally demanding. In this paper, we present an approximation algorithm for evaluating the TGOSPA metric, based on casting the TGOSPA problem as an unbalanced multimarginal optimal transport problem. Following recent advances in computational optimal transport, we introduce an entropy regularization and derive an iterative scheme for solving the Lagrangian dual of the regularized problem. Numerical results suggest that our proposed algorithm is more computationally efficient than the alternative of computing the exact metric using a linear programming solver, while still providing an adequate approximation of the metric.

Fast computation of the TGOSPA metric for multiple target tracking via unbalanced optimal transport

TL;DR

The paper tackles the computational bottleneck of evaluating the trajectory generalized optimal sub-pattern assignment (TGOSPA) metric in multi-target tracking by reformulating the LP-relaxed TGOSPA as an entropy-regularized, unbalanced multimarginal OT problem. It derives a Sinkhorn-type algorithm via block coordinate ascent in the Lagrangian dual and shows how the cost structure enables efficient projection computations with complexity per update, yielding linear convergence. Numerical results demonstrate substantial speed-ups over exact LP solvers while maintaining accurate approximations (≈1% relative error on larger problems), with potential for GPU acceleration. The approach provides a scalable, principled metric computation for large MTT scenarios and suggests future work on differentiable implementations for gradient-based data-driven tracking.

Abstract

In multiple target tracking, it is important to be able to evaluate the performance of different tracking algorithms. The trajectory generalized optimal sub-pattern assignment metric (TGOSPA) is a recently proposed metric for such evaluations. The TGOSPA metric is computed as the solution to an optimization problem, but for large tracking scenarios, solving this problem becomes computationally demanding. In this paper, we present an approximation algorithm for evaluating the TGOSPA metric, based on casting the TGOSPA problem as an unbalanced multimarginal optimal transport problem. Following recent advances in computational optimal transport, we introduce an entropy regularization and derive an iterative scheme for solving the Lagrangian dual of the regularized problem. Numerical results suggest that our proposed algorithm is more computationally efficient than the alternative of computing the exact metric using a linear programming solver, while still providing an adequate approximation of the metric.

Paper Structure

This paper contains 12 sections, 4 theorems, 25 equations, 3 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Background
  3. TGOSPA
  4. Multimarginal Optimal Transport
  5. A Novel Algorithm for Approximation of the Relaxed TGOSPA Metric
  6. TGOSPA as Multimarginal Optimal Transport
  7. Block Coordinate Ascent in the Lagrangian Dual
  8. Efficient Computation of Projections of Transport Tensors and Convergence of the Sinkhorn Algorithm
  9. Numerical Results
  10. Illustrating Convergence and Effect of Regularization
  11. Computational Efficiency
  12. Conclusions

Key Result

Theorem 1

Let $\bar{\mu} \in \mathbb{R}^{m + 1}$ and $\Tilde{\mu} \in \mathbb{R}^{n + 1}$ be defined by $\bar{\mu} = \left( \bm{1}_m^\top, n \right)^\top$ and $\tilde{\mu} = \left( \bm{1}_n^\top, m \right)^\top$, where $\bm{1}_\ell \in \mathbb{R}^\ell$ is a vector with all elements equal to one. Further, let where $\delta$ denotes the Kronecker delta, i.e. $\delta_{i,j} = 1$ if $i = j$ and $0$ otherwise. T

Figures (3)

  • Figure 1: Illustration of the problem structure. The top row shows the high level structure, the middle row the assignment matrices, and the bottom row the flows between rows in the assignment matrices.
  • Figure 2: Left: Simulated data. GT is the simulated ground truth data, and E is the simulated estimated trajectories. Middle: Convergence plot of Algorithm \ref{['alg:sinkhorn_t_gospa']} on the data in Figure \ref{['fig:misc']} (left). Right: Relative error depending on the parameter $\eta$.
  • Figure 3: Average runtimes and errors for Algorithm \ref{['alg:sinkhorn_t_gospa']} on varying data sizes. The top row shows results for fixed number of time steps $T$ and varying number of targets $m$. The bottom row shows results for varying number of time steps $T$ and fixed number of targets $m$. The shaded regions in the right column is one standard deviation from the mean.

Theorems & Definitions (11)

  • Remark 1
  • Theorem 1
  • proof
  • Remark 2
  • Remark 3
  • Proposition 1
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • ...and 1 more