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Tensor Decompositions for Online Grid-Based Terrain-Aided Navigation

J. Matoušek, J. Krejčí, J. Duník, R. Zanetti

TL;DR

This work presents a CPD-based grid filter for high-dimensional state estimation where dynamics are invertible and measurements are highly nonlinear. By encoding the state density as a CPD and performing both measurement and time updates in CPD form, the method achieves real-time performance even in 4D TAN-like problems, thanks to block-diagonal dynamics and sparsely coupled measurements. Key contributions include a Hadamard product in CPD, Lagrangian advection-diffusion updates, and rank-management via cp_als, enabling scalability beyond traditional grid-based filters. Empirical results on terrain-aided navigation show competitive accuracy with significantly reduced computation time, while outlining future work to relax diagonal noise assumptions and extend to more general dynamics.

Abstract

This paper presents a practical and scalable grid-based state estimation method for high-dimensional models with invertible linear dynamics and with highly non-linear measurements, such as the nearly constant velocity model with measurements of e.g. altitude, bearing, and/or range. Unlike previous tensor decomposition-based approaches, which have largely remained at the proof-of-concept stage, the proposed method delivers an efficient and practical solution by exploiting decomposable model structure-specifically, block-diagonal dynamics and sparsely coupled measurement dimensions. The algorithm integrates a Lagrangian formulation for the time update and leverages low-rank tensor decompositions to compactly represent and effectively propagate state densities. This enables real-time estimation for models with large state dimension, significantly extending the practical reach of grid-based filters beyond their traditional low-dimensional use. Although demonstrated in the context of terrain-aided navigation, the method is applicable to a wide range of models with decomposable structure. The computational complexity and estimation accuracy depend on the specific structure of the model. All experiments are fully reproducible, with source code provided alongside this paper (GitHub link: https://github.com/pesslovany/Matlab-LagrangianPMF).

Tensor Decompositions for Online Grid-Based Terrain-Aided Navigation

TL;DR

This work presents a CPD-based grid filter for high-dimensional state estimation where dynamics are invertible and measurements are highly nonlinear. By encoding the state density as a CPD and performing both measurement and time updates in CPD form, the method achieves real-time performance even in 4D TAN-like problems, thanks to block-diagonal dynamics and sparsely coupled measurements. Key contributions include a Hadamard product in CPD, Lagrangian advection-diffusion updates, and rank-management via cp_als, enabling scalability beyond traditional grid-based filters. Empirical results on terrain-aided navigation show competitive accuracy with significantly reduced computation time, while outlining future work to relax diagonal noise assumptions and extend to more general dynamics.

Abstract

This paper presents a practical and scalable grid-based state estimation method for high-dimensional models with invertible linear dynamics and with highly non-linear measurements, such as the nearly constant velocity model with measurements of e.g. altitude, bearing, and/or range. Unlike previous tensor decomposition-based approaches, which have largely remained at the proof-of-concept stage, the proposed method delivers an efficient and practical solution by exploiting decomposable model structure-specifically, block-diagonal dynamics and sparsely coupled measurement dimensions. The algorithm integrates a Lagrangian formulation for the time update and leverages low-rank tensor decompositions to compactly represent and effectively propagate state densities. This enables real-time estimation for models with large state dimension, significantly extending the practical reach of grid-based filters beyond their traditional low-dimensional use. Although demonstrated in the context of terrain-aided navigation, the method is applicable to a wide range of models with decomposable structure. The computational complexity and estimation accuracy depend on the specific structure of the model. All experiments are fully reproducible, with source code provided alongside this paper (GitHub link: https://github.com/pesslovany/Matlab-LagrangianPMF).
Paper Structure (19 sections, 1 theorem, 34 equations, 2 figures, 1 table)

This paper contains 19 sections, 1 theorem, 34 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Grid-based Bayesian Estimation
  3. Bayesian Recursive Relations
  4. Point - Mass Density
  5. Lagrangian Grid-based Solution
  6. Measurement update
  7. Time-Update
  8. Canonical Polyadic Decomposition
  9. Hadamard Product for the CPD Format
  10. Special Cases of Decomposition
  11. Tensor of Initial Condition Weights
  12. Tensor with Some Invariant Modes
  13. High-Dimensional Grid Based Filtering
  14. Measurement update
  15. Time-update
  16. ...and 4 more sections

Key Result

Lemma 1

The Hadamard product $C=A \odot B$ is given by Proof: Evaluating the product $C=A \odot B$ at the index $(i_1, \dots, i_D)$ reads whose tensor notation yields eq:CPDhadam. $\square$

Figures (2)

  • Figure 1: Points-mass density
  • Figure 2: Illustration of different grids and their projections for $x$ world direction at time step $k$.

Theorems & Definitions (1)

  • Lemma 1